CA Geometry: Compass construction

WEBVTT
Kind: captions
Language: en

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We are on problem 56.

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Scott is constructing a line
perpendicular to line

00:00:07.370 --> 00:00:09.580
L from point P.

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Fair enough.

00:00:10.650 --> 00:00:14.170
Which of the following should
be his first step?

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So he wants to draw
a line that looks

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something like this.

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He want to draw it
going straight up

00:00:18.970 --> 00:00:20.660
through the point P.

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And so how do you do that?

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Obviously, if you just use a
ruler, maybe by accident, you

00:00:23.860 --> 00:00:25.810
draw it at a slight angle or
something like that, right?

00:00:25.810 --> 00:00:28.890
So he wants to have
an exact line.

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Let's see, I don't even
know what he's doing

00:00:30.400 --> 00:00:31.120
in this first one.

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He's drawing these x's.

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I don't know how he's
determining

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where those x's go.

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So that A doesn't look right.

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Step B, it looks like he kind
of picked two points and is

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drawing and using his compass to
pivot off of those to kind

00:00:48.590 --> 00:00:50.480
of draw arcs there.

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But that's still not clear
how it could help.

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If he knew that he could do some
math and try to figure

00:00:55.840 --> 00:00:58.500
out that these are equidistant
from P.

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But still, what he wants to do
is find a point right around

00:01:01.770 --> 00:01:06.040
there, right below P, where if
he draws a line between P and

00:01:06.040 --> 00:01:07.870
that point, it'll be
perpendicular.

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So that doesn't seem
to be of much help.

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Here, he picked a point and he's
drawing an arc, but that

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arc doesn't really tell
me much information.

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Let me see, D.

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This looks interesting.

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So it looks like he took the
pivot of his compass and he

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drew a circle around with
a constant radius.

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Obviously, that's what
makes it a circle.

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And then what do he
could do now is he

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could take his pivot.

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He could you could mark each
of these points, right?

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If he marked each of those
points and just pivoted around

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them and drew a circle, he would
actually-- so let's say

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that around that point, he were
to pivot and the circles

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looks something like--
I don't know.

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And he would want to change the
radius a little bit, so

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let's say it looks something
like that.

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I'm going to try my
best to draw it.

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Let's say it looks something
like that.

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And then around that point,
he does the same thing.

00:02:03.070 --> 00:02:05.900
He pivots around it.

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I guess the two would have to
be big enough to intersect

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with each other.

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I know I'm drawing it
really horrible.

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The point at which they do
intersect would be equidistant

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between those two points.

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Another way to think about it,
it would be another point

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that's equidistant between the
two points, because when you

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do it first with P and you
draw that circle, you're

00:02:27.210 --> 00:02:29.230
saying both of these points
are going to be

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equidistant from P.

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Just by definition, right?

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This is a circle and that's
a constant radius.

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And then if you were to take
each of those points and draw

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circles or draw arcs-- so let's
say from that on you

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draw an arc like that and from
that one you draw an arc like

00:02:42.920 --> 00:02:46.790
that-- you'd say, wow, this
point is also going to be of a

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constant distance from
both of them.

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So if I were to draw a line
between both of these points

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using a straight edge,
that line will be

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perpendicular to line L.

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So if I were to just do
that, then it would be

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perpendicular, so I think
D is the first step.

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All right, problem 57: Which
triangle can be constructed

00:03:07.630 --> 00:03:09.950
using the following steps?

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OK, this is interesting.

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A lot of compass work here.

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Put the tip of the compass
on point A.

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Open the compass so that the
pencil tip is on point B.

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Draw arc above AB.

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So then they drew this
arc right here.

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That's this thing that I'm
trying to color in.

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Fair enough.

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Without changing the opening,
put the metal tip on points B,

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so now you put the pivot
there, and draw an arc

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interesting the first
point at C.

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So now they want us to draw that
second-- let me do it in

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another color.

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They're going to draw
that second arc.

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OK, now draw AC and BC.

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Now what have we drawn?

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So when you draw this first
arc-- what they drew is really

00:03:59.900 --> 00:04:02.970
a semicircle-- the radius
is constant.

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So if the radius is constant,
you know that this distance

00:04:06.570 --> 00:04:10.710
right here is going to be
equal to this distance.

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They're just both
radiuses of this

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semicircle or of this arc.

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They're just radiuses.

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They're equal to the length of
the opening of our compass.

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So that's going to
be equal to that.

00:04:21.050 --> 00:04:23.760
And then when you put the pivot
here and you keep that

00:04:23.760 --> 00:04:28.110
distance the same, so now the
pencil edge goes here, the

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distance is still there.

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And now when you do this arc,
you now know that this length

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is equal to this length, because
now they're both

00:04:34.840 --> 00:04:39.030
radiuses of this second arc.

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So now, you know all three sides
are equal, so this is an

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equilateral triangle.

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Equilateral, D.

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OK, the diagram shows
triangle ABC.

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Fair enough.

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Which statement would prove that
ABC is a right triangle?

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This is clearly not
a right angle.

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This is probably the right angle
for the right triangle.

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And this is something that maybe
you learned in algebra

00:05:10.550 --> 00:05:11.610
class, and if you didn't,
you're about to

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learn it right now.

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And I'm looking at
the choices.

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They talk a lot about slope.

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If I just have a line let's say
here, and it has slope M,

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and I want to say what is the
slope on a line that is

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perpendicular to this
line right here?

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Well then, it would
look like that.

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It would be perpendicular.

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It would have a 90-degree angle,
and its slope would be

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the negative inverse.

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The negative inverse of
this first slope.

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So if the slope from A to B is
a negative inverse of the

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slope for B to C, then
we're in business.

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These are definitely
perpendicular, at least line

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segments, and this would
be a 90-degree angle.

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Let's see.

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So what did I say?

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Slope AB, or we could say slope
BC, should be equal to

00:06:12.110 --> 00:06:22.050
the negative inverse of
the slope of A to B.

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See, if you multiply both sides
of that times the slope

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AB, you get slope AB
times slope BC is

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equal to negative 1.

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I just multiplied both sides
times slope of AB.

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And if we could go here, choice
B is exactly what we

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wrote right there.

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Next problem, 59:
Figure ABC, oh!

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It's a parallelogram.

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Fair enough.

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That's parallel to that.

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That's parallel to that.

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What are they asking us?

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What are the coordinates of the
point of intersection of

00:07:04.200 --> 00:07:05.240
the diagonals?

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So what are these coordinates?

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And we've mentioned it before,
but the big kind of crux of

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this problem or what you need
to know is that for any

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parallelogram, the diagonals
bisect each other.

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So that means that the distance
from here to here,

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from O to the intersection
point is equal to the

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intersection point to B.

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And so this intersection
point is the midpoint.

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It is bisected by line AC.

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And similarly, you can make the
argument that this line

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segment right here is equal
to-- is congruent

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to that line segment.

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If this is really the midpoint
between O and B, then we just

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have to find the midpoint
of their coordinates.

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And the way to find the midpoint
of two coordinates is

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actually very intuitive.

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You just average the
coordinates.

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So if I were to average the
x's-- so the x-coordinate here

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is going to be-- the
x-coordinate of point B, which

00:08:15.490 --> 00:08:19.340
is A plus C, plus the
x-coordinate of the origin

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which is 0 over 2,
because I had two

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points that I'm averaging.

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So that's going to be
the x-coordinate.

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And then y-coordinate is going
to be the y-coordinate

00:08:28.880 --> 00:08:34.039
coordinate of B plus the
y-coordinate of the origin.

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I'm just averaging them.

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The y-coordinate of the origin
is 0 divided by 2.

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This will give me the coordinate
of the midpoint

00:08:40.390 --> 00:08:42.950
between origin and B.

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So that equals A plus C over 2
and then B over 2, so this is

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A plus C over 2.

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That makes sense because A plus
C is going to be here

00:08:57.290 --> 00:09:00.710
someplace, and we just took the
average of the two between

00:09:00.710 --> 00:09:02.680
zero and that, and you
got the midpoint.

00:09:02.680 --> 00:09:06.920
And then the y-coordinate is
going to be B over 2, which

00:09:06.920 --> 00:09:08.970
makes sense because that up
there is B and we're just

00:09:08.970 --> 00:09:13.045
halfway between B and zero.

00:09:13.045 --> 00:09:15.800
So it's A plus C over
2, B over 2.

00:09:15.800 --> 00:09:17.070
And is that one of
the choices?

00:09:17.070 --> 00:09:22.210
A plus C over 2, I'm assuming
that they want us to do choice

00:09:22.210 --> 00:09:25.140
C and they just forgot to
type the B in there.

00:09:25.140 --> 00:09:27.020
This is supposed to be a B.

00:09:27.020 --> 00:09:28.170
Because this is definitely
not right.

00:09:28.170 --> 00:09:29.100
That's not right.

00:09:29.100 --> 00:09:31.620
And we know that this is right,
but it's not A plus

00:09:31.620 --> 00:09:32.890
something over on this side.

00:09:32.890 --> 00:09:34.610
It's just B over 2 on
the y-coordinate.

00:09:34.610 --> 00:09:35.880
So that's not right.

00:09:35.880 --> 00:09:37.270
All right, problem 60.

00:09:37.270 --> 00:09:40.580
I tried to squeeze it in.

00:09:40.580 --> 00:09:42.290
I don't know if you can
see the whole problem.

00:09:42.290 --> 00:09:47.300
But it says what type of
triangle is formed by the

00:09:47.300 --> 00:09:53.560
points A is 4 comma 2, B is 6
comma negative 1, and C is

00:09:53.560 --> 00:09:55.460
negative 1 comma 3?

00:09:55.460 --> 00:09:58.030
So I think the best thing is to
just try to graph it and at

00:09:58.030 --> 00:09:59.490
least start getting an
intuition, and then we can see

00:09:59.490 --> 00:10:02.970
the distances between the
points, and hopefully, figure

00:10:02.970 --> 00:10:05.510
out what type of
triangle it is.

00:10:05.510 --> 00:10:06.720
So let's see, some
of the points get

00:10:06.720 --> 00:10:09.090
a little bit negative.

00:10:09.090 --> 00:10:11.080
I'll have to draw some of
the negative quadrants.

00:10:13.880 --> 00:10:17.320
So if I were to draw
it like that.

00:10:17.320 --> 00:10:20.120
OK, let's see.

00:10:20.120 --> 00:10:21.570
So 4 comma 2.

00:10:30.490 --> 00:10:31.340
That's right there.

00:10:31.340 --> 00:10:33.340
That's point A.

00:10:33.340 --> 00:10:35.150
And then I have 6 comma
negative 1.

00:10:42.860 --> 00:10:44.500
That's point B.

00:10:44.500 --> 00:10:46.060
I don't know if you can
see it down there.

00:10:46.060 --> 00:10:48.320
And C is negative 1 comma 3.

00:10:55.823 --> 00:10:57.510
So it's out here.

00:11:01.260 --> 00:11:04.290
Now let me connect the dots.

00:11:04.290 --> 00:11:08.700
That's one side, that's another
side, and that's the

00:11:08.700 --> 00:11:09.290
other side.

00:11:09.290 --> 00:11:11.450
So off the bat, I just
know this isn't going

00:11:11.450 --> 00:11:12.420
to be a right triangle.

00:11:12.420 --> 00:11:13.990
It's not an equilateral
triangle.

00:11:13.990 --> 00:11:17.455
And the only way it's going to
be an isosceles triangle is if

00:11:17.455 --> 00:11:20.600
this length is equal to that
length, so let's just try it.

00:11:20.600 --> 00:11:22.280
Let's just test it out.

00:11:22.280 --> 00:11:25.170
So what is the distance
from A to C?

00:11:30.490 --> 00:11:35.240
The distance squared from A to
C is equal to the differences

00:11:35.240 --> 00:11:36.630
in their x's.

00:11:36.630 --> 00:11:40.060
So 4 minus negative
1, so there's a

00:11:40.060 --> 00:11:41.905
difference of 5, right?

00:11:41.905 --> 00:11:46.260
So there's a difference in
their x's squared plus a

00:11:46.260 --> 00:11:48.140
difference in their y's.

00:11:48.140 --> 00:11:50.910
So 2 and 3.

00:11:50.910 --> 00:11:53.170
You could just say 2 minus
3 or 3 minus is 2.

00:11:53.170 --> 00:11:53.580
It doesn't matter.

00:11:53.580 --> 00:11:56.690
We just care about
the difference.

00:11:56.690 --> 00:11:58.050
Plus one squared.

00:11:58.050 --> 00:12:02.080
So the distance squared
is equal to 25 plus

00:12:02.080 --> 00:12:04.540
1 is equal to 26.

00:12:04.540 --> 00:12:07.580
So this distance is the
square root of 26.

00:12:07.580 --> 00:12:10.630
And the distance between
A and B, same logic.

00:12:10.630 --> 00:12:13.720
Let's see, when you go from
the distance squared, the

00:12:13.720 --> 00:12:16.210
difference is in there x's.

00:12:16.210 --> 00:12:18.740
Between 6 and 4, you have a
distance of 2, so it's 2

00:12:18.740 --> 00:12:21.810
squared plus the difference
in their y's.

00:12:21.810 --> 00:12:24.770
2 and negative 1 are
3 apart, right?

00:12:24.770 --> 00:12:27.990
Plus 3 squared.

00:12:27.990 --> 00:12:33.486
And so that is equal to 4 plus
9, which is equal to 13, so

00:12:33.486 --> 00:12:36.090
this is equal to the
square root of 13.

00:12:36.090 --> 00:12:38.480
OK, and this number down here,
you can figure it out, but

00:12:38.480 --> 00:12:40.570
it's going to be bigger than
both of them, right?

00:12:40.570 --> 00:12:42.260
You can just look at
it and say that.

00:12:42.260 --> 00:12:44.040
So this is definitely
a scalene triangle.

00:12:44.040 --> 00:12:46.550
All of the sides
are different.

00:12:46.550 --> 00:12:49.020
Anyway, see you in
the next video.


CA Geometry: Pythagorean theorem, compass constructions

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.550
We're on problem 51.

00:00:02.550 --> 00:00:05.970
And they say, a diagram from
a proof of the Pythagorean

00:00:05.970 --> 00:00:07.890
theorem is pictured below.

00:00:07.890 --> 00:00:11.150
And they they say, which
statement would not be used in

00:00:11.150 --> 00:00:13.310
the proof of the Pythagorean
theorem?

00:00:13.310 --> 00:00:15.620
So since they have drawn this
diagram out, I think we might

00:00:15.620 --> 00:00:18.320
as well just kind of do the
proof and then we can look at

00:00:18.320 --> 00:00:20.290
their choices and see which
ones kind of match

00:00:20.290 --> 00:00:21.890
up to what we did.

00:00:21.890 --> 00:00:24.380
Hopefully, they do
it the same way.

00:00:24.380 --> 00:00:26.140
And this is a pretty
neat proof of

00:00:26.140 --> 00:00:27.530
the Pythagorean theorem.

00:00:27.530 --> 00:00:28.360
I don't think I've
done it yet.

00:00:28.360 --> 00:00:29.610
So I might as well do it now.

00:00:32.850 --> 00:00:35.130
Well, let's figure out
what the area of this

00:00:35.130 --> 00:00:37.040
large square is right.

00:00:37.040 --> 00:00:38.580
Well there's two ways to think
about it you could just say,

00:00:38.580 --> 00:00:39.760
OK, this is a square.

00:00:39.760 --> 00:00:41.250
That's a, that's b.

00:00:41.250 --> 00:00:43.300
Well this is going
to be b as well.

00:00:43.300 --> 00:00:45.320
This is going to be a as well.

00:00:45.320 --> 00:00:48.480
So the area of the square is
going to be the length of one

00:00:48.480 --> 00:00:51.450
of its sides squared.

00:00:51.450 --> 00:00:55.230
So we could say the
whole square's

00:00:55.230 --> 00:00:57.470
area is a plus b squared.

00:01:01.020 --> 00:01:07.960
And that's equal to a squared
plus 2ab plus b squared.

00:01:07.960 --> 00:01:08.780
Fair enough.

00:01:08.780 --> 00:01:12.892
Now we can also say that the
area of this larger square,

00:01:12.892 --> 00:01:15.090
and it's a bit of an optical
illusion, it looks like it's

00:01:15.090 --> 00:01:16.910
tilted to the left because
of the way it's drawn.

00:01:16.910 --> 00:01:19.800
But anyway, that the area of
this larger square is also the

00:01:19.800 --> 00:01:24.920
area of these four triangles
plus the area of

00:01:24.920 --> 00:01:26.560
this smaller square.

00:01:26.560 --> 00:01:28.580
So this, the area of the
larger square, which we

00:01:28.580 --> 00:01:31.720
figured out just by taking one
side of it and squaring it,

00:01:31.720 --> 00:01:33.700
that should be equal
to the area of the

00:01:33.700 --> 00:01:35.190
four smaller triangles.

00:01:35.190 --> 00:01:37.200
So there's four of them.

00:01:37.200 --> 00:01:39.670
And what's the area
of each of them.

00:01:39.670 --> 00:01:42.130
Let's see, let's just
pick this one.

00:01:42.130 --> 00:01:43.640
1/2 base times height.

00:01:43.640 --> 00:01:46.410
So it's 1/2 times a times b.

00:01:51.770 --> 00:01:54.710
So 1/2 ab is one of these and
I multiply by 4 to get all

00:01:54.710 --> 00:01:56.980
four of these triangles.

00:01:56.980 --> 00:02:00.970
And then we want to add the area
of this inside square.

00:02:00.970 --> 00:02:04.820
And that's just going
to be c squared.

00:02:04.820 --> 00:02:08.090
So plus c squared.

00:02:08.090 --> 00:02:10.150
Let's see if we can
simplify this.

00:02:10.150 --> 00:02:19.400
So you get a squared plus 2ab
plus b squared is equal to 4

00:02:19.400 --> 00:02:25.440
times 1/2 is 2ab
plus c squared.

00:02:25.440 --> 00:02:27.390
Well, we could subtract
2ab from both

00:02:27.390 --> 00:02:29.820
sides of this equation.

00:02:29.820 --> 00:02:31.360
The top and the bottom of
this equation the way

00:02:31.360 --> 00:02:32.460
I've written it.

00:02:32.460 --> 00:02:36.750
But if we do that, subtract 2ab
from there, subtract 2ab

00:02:36.750 --> 00:02:41.380
from there, and you're left with
a squared plus b squared

00:02:41.380 --> 00:02:44.680
is equal to c squared, which
is the Pythagorean theorem.

00:02:44.680 --> 00:02:46.170
And we've proved it.

00:02:46.170 --> 00:02:50.100
So let's see which of their
choices matches what we did.

00:02:50.100 --> 00:02:53.170
OK, which statement would not
be used in the proof of the

00:02:53.170 --> 00:02:54.550
Pythagorean theorem.

00:02:54.550 --> 00:02:57.460
The area of a triangle
equals 1/2 ab.

00:02:57.460 --> 00:02:59.720
We used that.

00:02:59.720 --> 00:03:03.060
The four right triangles
are congruent.

00:03:03.060 --> 00:03:05.650
No, we used that.

00:03:05.650 --> 00:03:09.850
The area of the inner square is
equal to half of the area

00:03:09.850 --> 00:03:11.220
of the larger square.

00:03:11.220 --> 00:03:12.260
We didn't use that.

00:03:12.260 --> 00:03:14.560
I think this is the
one that would not

00:03:14.560 --> 00:03:15.330
be used in the proof.

00:03:15.330 --> 00:03:18.500
Choice D, the area of the larger
square is equal to the

00:03:18.500 --> 00:03:21.790
sum of the squares of the
smaller square and the four

00:03:21.790 --> 00:03:22.260
congruent triangles.

00:03:22.260 --> 00:03:24.380
No, that that was the
crux of the proof.

00:03:24.380 --> 00:03:25.280
So we definitely used that.

00:03:25.280 --> 00:03:26.180
So C is our answer.

00:03:26.180 --> 00:03:27.980
That's the statement
that would not

00:03:27.980 --> 00:03:29.230
be used in the proof.

00:03:32.080 --> 00:03:34.910
I'm learning to copy and
paste ahead of time.

00:03:34.910 --> 00:03:36.640
So I don't waste your time.

00:03:36.640 --> 00:03:40.410
All right, a right triangle's
hypotenuse has length 5.

00:03:40.410 --> 00:03:42.570
If one leg has length 2,
what is the length

00:03:42.570 --> 00:03:43.820
of the other leg?

00:03:50.230 --> 00:03:55.390
Pythagorean theorem, x squared
plus 2 squared is equal to 5

00:03:55.390 --> 00:03:58.490
squared, because 5 is
the hypotenuse.

00:03:58.490 --> 00:04:02.810
x squared plus 4
is equal to 25.

00:04:02.810 --> 00:04:04.660
Subtract 4 from both sides.

00:04:04.660 --> 00:04:07.610
x squared is equal to 21.

00:04:07.610 --> 00:04:12.080
So x is equal to the
square root of 21.

00:04:12.080 --> 00:04:14.560
So choice B.

00:04:14.560 --> 00:04:16.959
Next question.

00:04:16.959 --> 00:04:21.089
A new pipeline is being
constructed to reroute oil

00:04:21.089 --> 00:04:26.430
flow around the exterior of a
national wildlife preserve.

00:04:26.430 --> 00:04:28.860
I guess that's the national
wildlife preserve.

00:04:28.860 --> 00:04:32.210
The plan showing the old
pipeline and the new route is

00:04:32.210 --> 00:04:33.820
shown below.

00:04:33.820 --> 00:04:39.000
OK, how many extra miles will
the oil flow once the new

00:04:39.000 --> 00:04:40.270
route is establised.

00:04:40.270 --> 00:04:44.570
So the new route is going to
be 60 miles plus 32 miles.

00:04:44.570 --> 00:04:48.060
So the new route is 92 miles.

00:04:48.060 --> 00:04:49.080
So what was the old route?

00:04:49.080 --> 00:04:53.360
Well the old route was the
hypotenuse of this triangle.

00:04:53.360 --> 00:04:57.250
So we could say, let's
call that x.

00:04:57.250 --> 00:05:10.590
60 squared plus 32 squared
is equal to x squared.

00:05:10.590 --> 00:05:12.150
Because that's the hypotenuse.

00:05:12.150 --> 00:05:18.450
And these numbers, that's a bit
of a pain to deal with.

00:05:18.450 --> 00:05:22.180
Maybe if I can factor out
something here I can make it

00:05:22.180 --> 00:05:22.980
more interesting.

00:05:22.980 --> 00:05:27.510
So I don't have to multiply out
60 squared and 32 squared

00:05:27.510 --> 00:05:31.260
and all of the rest.
Well, let me see.

00:05:34.040 --> 00:05:37.175
Both of those are
divisible by 4.

00:05:41.400 --> 00:05:44.260
So then I would have 15 and 8.

00:05:44.260 --> 00:05:48.290
Yeah, that still doesn't
make it that useful.

00:05:48.290 --> 00:05:49.270
So I'll just multiply
them out.

00:05:49.270 --> 00:05:53.090
So this is 3600.

00:05:53.090 --> 00:05:56.700
At 32 squared, let's
see, 32 times 32.

00:05:56.700 --> 00:05:58.750
2 times 32 is 64.

00:05:58.750 --> 00:06:02.200
3 times 2 is 6.

00:06:02.200 --> 00:06:04.730
3 times 3 is 9.

00:06:04.730 --> 00:06:08.770
So it's 1024.

00:06:08.770 --> 00:06:13.360
Plus 1024 is equal
to x squared.

00:06:13.360 --> 00:06:15.120
So let me just switch
both sides.

00:06:15.120 --> 00:06:23.040
x squared is equal to 3600
plus 1024 is 4624.

00:06:23.040 --> 00:06:24.413
Let me see if I can get
an approximate.

00:06:26.980 --> 00:06:32.260
So x is going to be the square
root of this thing right here.

00:06:32.260 --> 00:06:34.840
So let's see if I can get
a handle at least on the

00:06:34.840 --> 00:06:36.940
magnitude of where
this would be.

00:06:36.940 --> 00:06:43.370
So 20 times 20 is 400.

00:06:43.370 --> 00:06:45.050
So this is way too small.

00:06:45.050 --> 00:06:48.740
60 times 60 is 3600.

00:06:48.740 --> 00:06:51.290
So 68 times 68, this
looks right.

00:06:51.290 --> 00:06:53.020
Especially because 8 times
8 should end in a 4.

00:06:53.020 --> 00:06:54.190
Let me try that out.

00:06:54.190 --> 00:06:57.250
68 times 68.

00:06:57.250 --> 00:07:00.130
8 times 8 is 64.

00:07:00.130 --> 00:07:07.370
8 times 6 is 48 plus 6 is 54.

00:07:07.370 --> 00:07:09.740
6 times 8, 48.

00:07:09.740 --> 00:07:16.680
6 times 6, 36 plus 4 is 40.

00:07:16.680 --> 00:07:18.300
4624.

00:07:18.300 --> 00:07:23.090
So x is 68.

00:07:23.090 --> 00:07:28.285
Oh, I used 68, I shouldn't have.
Because they don't want

00:07:28.285 --> 00:07:30.820
to know how long was
the old pipeline.

00:07:30.820 --> 00:07:31.760
That's 68.

00:07:31.760 --> 00:07:33.570
It just happened to be
one of the choices.

00:07:33.570 --> 00:07:35.770
That's just to make
sure that you read

00:07:35.770 --> 00:07:36.540
the question properly.

00:07:36.540 --> 00:07:38.740
But they want to know how
much longer is going

00:07:38.740 --> 00:07:41.680
to be the new pipeline.

00:07:41.680 --> 00:07:43.870
So the new one was 92.

00:07:43.870 --> 00:07:45.800
And the old one is 68.

00:07:45.800 --> 00:07:47.150
Good thing they had that
number there so I

00:07:47.150 --> 00:07:48.260
could try it out.

00:07:48.260 --> 00:07:50.690
That was the square
root of 4624.

00:07:50.690 --> 00:07:52.690
So how much longer
is the new one?

00:07:52.690 --> 00:08:05.200
Well 92 minus 68 that's
24 miles.

00:08:05.200 --> 00:08:06.990
So choice A.

00:08:06.990 --> 00:08:07.660
Not choice B.

00:08:07.660 --> 00:08:09.980
B is how long the old
pipeline was.

00:08:09.980 --> 00:08:13.500
We want to know how much longer
the new route is.

00:08:13.500 --> 00:08:14.500
That was tricky.

00:08:14.500 --> 00:08:19.200
Well not tricky, but I kind of
fell for it by forgetting what

00:08:19.200 --> 00:08:20.280
the question was about.

00:08:20.280 --> 00:08:22.040
Anyway, next question.

00:08:22.040 --> 00:08:25.000
Marcia is using a straightedge
and compass to do the

00:08:25.000 --> 00:08:26.790
construction below.

00:08:26.790 --> 00:08:28.230
Interesting.

00:08:28.230 --> 00:08:31.330
Which best describes the
construction Marcia is doing.

00:08:31.330 --> 00:08:34.330
So, I assume when they
say construction

00:08:34.330 --> 00:08:36.690
she's drawing something.

00:08:36.690 --> 00:08:37.500
Let's see what it looks like.

00:08:37.500 --> 00:08:40.230
It looks she's taking her
compass, she's probably

00:08:40.230 --> 00:08:42.220
putting one of the points
here, she put one of the

00:08:42.220 --> 00:08:45.220
points there and then she
kind of drew this arc.

00:08:45.220 --> 00:08:47.420
And then it looks like she put
the point there and then she

00:08:47.420 --> 00:08:49.110
drew that arc.

00:08:49.110 --> 00:08:53.370
And then she put the point
here and drew that arc.

00:08:53.370 --> 00:08:58.110
And then put the point there
and drew that arc.

00:08:58.110 --> 00:09:02.050
And the end result, it seems
like the reason why she picked

00:09:02.050 --> 00:09:05.580
this point here is it goes
through this line L.

00:09:05.580 --> 00:09:08.950
So she's probably trying to find
another point here, so

00:09:08.950 --> 00:09:10.290
that she can draw
another line.

00:09:10.290 --> 00:09:11.600
Because they say she
has a straightedge.

00:09:11.600 --> 00:09:13.060
A straightedge is to
draw these lines.

00:09:13.060 --> 00:09:14.850
A compass is to draw
these curves.

00:09:14.850 --> 00:09:20.800
So if she were to draw another
line between these two points,

00:09:20.800 --> 00:09:26.970
it looks something like
that, then she would

00:09:26.970 --> 00:09:28.320
have parallel lines.

00:09:28.320 --> 00:09:29.960
The reason why she would have
parallel lines is because

00:09:29.960 --> 00:09:31.640
these would be corresponding
angles and

00:09:31.640 --> 00:09:32.900
they would be congruent.

00:09:32.900 --> 00:09:35.020
And so if you have a transversal
the corresponding

00:09:35.020 --> 00:09:37.700
angles are congruent, you're
dealing with parallel lines.

00:09:37.700 --> 00:09:40.930
So my read of this question is
that she's probably trying to

00:09:40.930 --> 00:09:45.630
draw a line that is
parallel to L.

00:09:45.630 --> 00:09:48.230
A line through P parallel
to line L.

00:09:48.230 --> 00:09:50.960
Yeah, that's what I think
she's trying to do.

00:09:50.960 --> 00:09:54.710
All right, choice A.

00:09:54.710 --> 00:09:56.820
55.

00:09:56.820 --> 00:09:59.525
Given angle A.

00:09:59.525 --> 00:10:01.700
So given this angle.

00:10:01.700 --> 00:10:04.870
What is the first step in
constructing the angle

00:10:04.870 --> 00:10:08.130
bisector of angle A?

00:10:14.460 --> 00:10:19.060
OK, well actually I've
never done this.

00:10:19.060 --> 00:10:21.000
But I can assume that
if I have a compass.

00:10:21.000 --> 00:10:22.900
You know what a compass is,
it has those two points.

00:10:22.900 --> 00:10:25.230
One of them is like
a pivot point.

00:10:25.230 --> 00:10:27.580
It looks something like this.

00:10:27.580 --> 00:10:31.330
It looks like it has a little
pivot point, and then on the

00:10:31.330 --> 00:10:35.400
other side you can stick
your pencil.

00:10:35.400 --> 00:10:38.040
And you can adjust it up here.

00:10:38.040 --> 00:10:39.870
And the bottom line, you pivot
around this and then you can

00:10:39.870 --> 00:10:44.130
draw circles of arbitrary
radiuses.

00:10:44.130 --> 00:10:46.260
It seems like that's
what they did here.

00:10:46.260 --> 00:10:50.140
So if I want to draw the angle
bisector of a, just thinking

00:10:50.140 --> 00:10:53.920
about it, it seems I could put
the pivot point here, and then

00:10:53.920 --> 00:10:57.930
I can put the pencil and
I can draw this circle.

00:10:57.930 --> 00:11:01.110
And really, as long as I just
find the two points that it

00:11:01.110 --> 00:11:04.160
intersects those two lines
or those two rays,

00:11:04.160 --> 00:11:04.810
then I'll be fine.

00:11:04.810 --> 00:11:05.950
And I could have done
it anywhere.

00:11:05.950 --> 00:11:07.830
I could have done it here.

00:11:07.830 --> 00:11:08.780
I could have done it out here.

00:11:08.780 --> 00:11:10.480
They just picked
points B and C.

00:11:10.480 --> 00:11:12.270
And then from each of those
points, you can

00:11:12.270 --> 00:11:15.090
put your pivot here.

00:11:15.090 --> 00:11:17.670
If you put your pivot here, and
then you were to draw a

00:11:17.670 --> 00:11:18.830
circle around that, you
would have gotten

00:11:18.830 --> 00:11:22.250
this one right here.

00:11:22.250 --> 00:11:24.560
And then if you were to put your
pivot point right here,

00:11:24.560 --> 00:11:27.080
draw a circle, you would
be able to draw that.

00:11:27.080 --> 00:11:31.000
And then where they interect,
that would that would give you

00:11:31.000 --> 00:11:33.620
an indication of where the
angle bisector is.

00:11:33.620 --> 00:11:37.240
And you could then draw that
line to where they intersect.

00:11:37.240 --> 00:11:39.530
So let's see, they say what is
the first step in constructing

00:11:39.530 --> 00:11:41.730
the angle bisector of angle a.

00:11:41.730 --> 00:11:44.030
So they say draw ray AD.

00:11:44.030 --> 00:11:46.640
Well that seems like that
would be the last step.

00:11:46.640 --> 00:11:47.280
Then you're done.

00:11:47.280 --> 00:11:50.330
Draw AD, that is the
angle biector.

00:11:50.330 --> 00:11:55.590
Draw a line segment connecting
points B and C.

00:11:55.590 --> 00:11:58.020
No, that's useless.

00:11:58.020 --> 00:11:59.320
You don't need a line segment.

00:11:59.320 --> 00:12:01.710
I mean even what they have
drawn, that's an arc.

00:12:01.710 --> 00:12:03.490
It's not a line.

00:12:03.490 --> 00:12:05.960
From points B and C, draw
equal arcs that

00:12:05.960 --> 00:12:07.280
intersect at D.

00:12:07.280 --> 00:12:08.750
That was the second step.

00:12:08.750 --> 00:12:10.960
You have to have points B and
C before you can draw those

00:12:10.960 --> 00:12:13.070
equal arcs.

00:12:13.070 --> 00:12:15.590
From point A, draw an arc that
intersects the side of the

00:12:15.590 --> 00:12:19.100
angle at points B and C.

00:12:19.100 --> 00:12:19.920
Yeah, that's what we said.

00:12:19.920 --> 00:12:20.780
That was the first step.

00:12:20.780 --> 00:12:24.090
Put your pivot here, and use
your pencil to draw the arc.

00:12:24.090 --> 00:12:25.720
You say OK, this point
and this point.

00:12:25.720 --> 00:12:27.710
So that would be
the first step.

00:12:27.710 --> 00:12:29.990
D.

00:12:29.990 --> 00:12:32.990
And I'm all out of problems
and I'm out of time.

00:12:32.990 --> 00:12:34.810
See you in the next video.


CA Geometry: Deducing angle measures

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.260
We're on problem 46.

00:00:03.260 --> 00:00:10.960
In the figure below, line
segment AB is parallel to C.

00:00:10.960 --> 00:00:12.010
I think there's a typo here.

00:00:12.010 --> 00:00:15.130
I've got to believe this
has got to be CD.

00:00:15.130 --> 00:00:16.850
That's got to be a typo.

00:00:16.850 --> 00:00:20.690
All right, so AB, I'm assuming,
is parallel to CD.

00:00:20.690 --> 00:00:22.580
That's probably what they wanted
to have, because you

00:00:22.580 --> 00:00:24.580
can't be parallel to a vertex.

00:00:24.580 --> 00:00:27.860
And they say, what is
the value of x?

00:00:27.860 --> 00:00:30.620
So the way to think about it is,
if these two are parallel

00:00:30.620 --> 00:00:33.080
lines, then this segment up
here is a transversal.

00:00:33.080 --> 00:00:36.500
If we just extend the two lines
like this, let me see if

00:00:36.500 --> 00:00:37.890
I can do it well.

00:00:37.890 --> 00:00:41.510
So if I extended that line like
that, and if I extended

00:00:41.510 --> 00:00:45.690
this line like that, then
segment AD is just a

00:00:45.690 --> 00:00:46.230
transversal.

00:00:46.230 --> 00:00:48.890
Let me see if I can
do that one right.

00:00:48.890 --> 00:00:51.610
So if I just extended it like
that, you see it's a

00:00:51.610 --> 00:00:55.110
transversal, and I could go the
other direction as well.

00:00:55.110 --> 00:00:56.860
I think you get the idea.

00:00:56.860 --> 00:00:58.810
So if that is a transversal,
what do you know about

00:00:58.810 --> 00:00:59.260
transversals?

00:00:59.260 --> 00:01:06.900
We know that this angle right
here, I'll draw it small, is

00:01:06.900 --> 00:01:08.920
congruent to this
angle over here.

00:01:08.920 --> 00:01:14.790
So the measure of this angle
is also x plus 40.

00:01:14.790 --> 00:01:16.650
Because they're corresponding
angles, and you could see that

00:01:16.650 --> 00:01:18.460
by inspection, and if you moved
around the transversal,

00:01:18.460 --> 00:01:20.400
it would make sense that
that's the case.

00:01:20.400 --> 00:01:22.800
So this is x plus 40 and this
is minus 40, and they're

00:01:22.800 --> 00:01:24.570
clearly supplements
of each other.

00:01:24.570 --> 00:01:25.940
They're supplementary angles.

00:01:25.940 --> 00:01:27.510
Then the sum of these
two angles have

00:01:27.510 --> 00:01:28.450
to be equal to 180.

00:01:28.450 --> 00:01:29.350
So let's figure it out.

00:01:29.350 --> 00:01:37.500
So x minus 40 plus x plus 40
is equal to 180, because

00:01:37.500 --> 00:01:39.420
they're supplementary.

00:01:39.420 --> 00:01:40.960
The 40's cancel out.

00:01:40.960 --> 00:01:43.840
So you just minus 40, plus
40 adds up to zero.

00:01:43.840 --> 00:01:47.420
So you're left with 2x
is equal to 180.

00:01:47.420 --> 00:01:49.920
x is equal to 90.

00:01:49.920 --> 00:01:51.170
So it's D.

00:01:54.650 --> 00:01:59.410
47: The measures of the interior
angles of a pentagon

00:01:59.410 --> 00:02:05.110
are 2x, 6x, 4x minus 6, 2x
minus 16, and 6x plus 2.

00:02:05.110 --> 00:02:08.990
What is the measure in degrees
of the largest angle?

00:02:08.990 --> 00:02:12.590
OK, so first of all, we have to
remember what is the sum of

00:02:12.590 --> 00:02:14.470
the interior angles
of a pentagon?

00:02:14.470 --> 00:02:17.160
And that's where I always draw
an arbitrary pentagon.

00:02:17.160 --> 00:02:20.260
Let me see if I can do that.

00:02:20.260 --> 00:02:22.760
Actually, there's a
polygon tool here.

00:02:22.760 --> 00:02:24.010
How does it work?

00:02:28.590 --> 00:02:29.840
I'm just trying to
draw a pentagon.

00:02:37.070 --> 00:02:39.130
I don't know if that's any
different than the line tool,

00:02:39.130 --> 00:02:39.750
but regardless.

00:02:39.750 --> 00:02:41.560
So how many triangles can
I draw in a pentagon?

00:02:41.560 --> 00:02:44.270
And that tells me what my total
interior angles are.

00:02:44.270 --> 00:02:46.970
And there is a formula for that,
but I like relying on

00:02:46.970 --> 00:02:50.300
your reasoning more than the
formula, because you might

00:02:50.300 --> 00:02:53.170
forget the formula, or even
worse, you might remember it,

00:02:53.170 --> 00:02:55.190
but not have the confidence to
use it, or you might remember

00:02:55.190 --> 00:02:56.930
it wrong ten years
in the future.

00:02:56.930 --> 00:02:58.540
So the best thing to do, if
you have a polygon, is to

00:02:58.540 --> 00:03:00.180
count the triangles in it.

00:03:00.180 --> 00:03:01.330
Straightforward enough.

00:03:01.330 --> 00:03:04.060
It's almost easier than
using the formula.

00:03:04.060 --> 00:03:06.810
So a pentagon has three
triangles in it.

00:03:06.810 --> 00:03:14.580
So the sum of its interior
angles are going to be 3 times

00:03:14.580 --> 00:03:18.170
180, because it has
3 triangles in it.

00:03:18.170 --> 00:03:20.070
Each triangle has 180 degrees.

00:03:20.070 --> 00:03:22.450
And I know you can't see
what I just wrote.

00:03:22.450 --> 00:03:24.710
So the sum of all of these
angles are going to be the sum

00:03:24.710 --> 00:03:27.620
of all of the angles in
all three triangles.

00:03:27.620 --> 00:03:31.260
So it's 3 times 180 is
equal to 540 degrees.

00:03:31.260 --> 00:03:34.340
So that's the sum of all of
these interior angles.

00:03:34.340 --> 00:03:37.400
Now, they say that each of them
are 2x, 6x, et cetera.

00:03:37.400 --> 00:03:41.570
So the sum of all of these terms
have to be equal 540.

00:03:41.570 --> 00:03:42.365
I'm going to write
them vertically.

00:03:42.365 --> 00:03:43.660
It makes them easier to add.

00:03:43.660 --> 00:03:53.810
So if we write there 2x, 6x,
4x minus 6, 2x minus

00:03:53.810 --> 00:03:59.320
16, and 6x plus 2.

00:03:59.320 --> 00:04:03.100
This is going to be the
largest, right?

00:04:03.100 --> 00:04:04.610
That sum is going
to equal 540.

00:04:04.610 --> 00:04:05.660
So let's add this up.

00:04:05.660 --> 00:04:09.050
Minus 6, minus 16,
that's minus 22.

00:04:09.050 --> 00:04:13.970
Plus 2 is minus 20.

00:04:13.970 --> 00:04:15.220
That's right.

00:04:18.240 --> 00:04:27.820
And 2x plus 6x that is 8x plus
4 is 12, 12 plus 2 is 14, 14

00:04:27.820 --> 00:04:30.280
plus 6 is 20.

00:04:30.280 --> 00:04:38.220
So we have 20x minus 20 is
equal to 540 degrees.

00:04:38.220 --> 00:04:38.990
Let me write it again.

00:04:38.990 --> 00:04:45.890
20x minus 20 is equal to 540.

00:04:45.890 --> 00:04:48.740
Let's divide both sides of
this equation by 20.

00:04:48.740 --> 00:04:56.800
So you get x 1 minus 1 is equal
to-- it would be 54

00:04:56.800 --> 00:05:02.610
divided by 2, which
is equal to 27.

00:05:02.610 --> 00:05:05.790
Add 1 to both sides,
x is equal to 28.

00:05:05.790 --> 00:05:08.040
And they want to know, what is
a measure in degrees of the

00:05:08.040 --> 00:05:08.620
largest angle?

00:05:08.620 --> 00:05:09.890
That's going to be this one.

00:05:09.890 --> 00:05:10.810
That's the largest one.

00:05:10.810 --> 00:05:12.900
It's 6 times x plus 2.

00:05:12.900 --> 00:05:21.320
So 6 times 28, that's 48.

00:05:21.320 --> 00:05:24.470
2 times 6 is 12 plus 4 is 168.

00:05:24.470 --> 00:05:29.510
So it's 168 plus 2.

00:05:29.510 --> 00:05:31.960
It's 170 degrees.

00:05:31.960 --> 00:05:34.610
Choice C.

00:05:34.610 --> 00:05:42.290
Problem 48: What is the
measure of angle 1?

00:05:42.290 --> 00:05:44.770
So this we're going into
the angle game.

00:05:44.770 --> 00:05:46.530
And these are fun, because
they are kind of these

00:05:46.530 --> 00:05:48.840
deductive reasoning problems
where you just use a couple of

00:05:48.840 --> 00:05:51.220
simple rules and just fill
in the whole thing.

00:05:51.220 --> 00:05:52.590
So let's think about it.

00:05:52.590 --> 00:05:53.680
This is 36 degrees.

00:05:53.680 --> 00:05:55.720
They tell us that this
whole angle right

00:05:55.720 --> 00:05:56.430
here is a right angle.

00:05:56.430 --> 00:05:59.560
So this angle right here is
going to be the complement to

00:05:59.560 --> 00:06:01.100
36 degrees.

00:06:01.100 --> 00:06:03.790
36 plus this angle have
to be equal to 90.

00:06:03.790 --> 00:06:05.270
So what's this one?

00:06:05.270 --> 00:06:11.860
This one is 90 minus
36, which is 54.

00:06:11.860 --> 00:06:13.980
That's going to be 54 degrees.

00:06:13.980 --> 00:06:15.560
90 minus 30 is 60.

00:06:15.560 --> 00:06:17.090
Right, that's 54.

00:06:17.090 --> 00:06:20.060
And this angle right here,
that's going to be the

00:06:20.060 --> 00:06:21.810
supplement of 88.

00:06:21.810 --> 00:06:25.130
So this is going to be-- I'll
do it in a different color--

00:06:25.130 --> 00:06:28.940
180 minus 88.

00:06:28.940 --> 00:06:33.260
That is equal to 92 degrees.

00:06:33.260 --> 00:06:37.290
Now this angle 1 plus the 54
plus the 92 is equal to 180.

00:06:37.290 --> 00:06:44.290
So we know that-- let's say
angle one plus 54 plus 92 is

00:06:44.290 --> 00:06:51.920
equal to 180.

00:06:51.920 --> 00:06:56.860
This is 146 is equal to 180.

00:06:56.860 --> 00:06:58.790
Subtract 146 from both sides.

00:06:58.790 --> 00:07:06.430
The measure of angle one is
equal to 80 minus 40 is 40, so

00:07:06.430 --> 00:07:09.820
80 minus 46 is equal
to 34 degrees.

00:07:09.820 --> 00:07:12.930
So the answer is A.

00:07:12.930 --> 00:07:20.460
Problem 49: What is the
measure of angle WZX?

00:07:20.460 --> 00:07:22.820
So they want to know what this
angle right here is.

00:07:22.820 --> 00:07:25.110
Let's do the angle
game some more.

00:07:25.110 --> 00:07:27.950
Let's see, we can immediately
figure out what this angle is,

00:07:27.950 --> 00:07:32.350
because it is the supplement
of 132 degrees, so this is

00:07:32.350 --> 00:07:40.460
going to be 180 minus 132.

00:07:40.460 --> 00:07:44.950
So this is 48 degrees.

00:07:44.950 --> 00:07:46.580
This angle plus this angle
is going to be

00:07:46.580 --> 00:07:49.820
equal to this angle.

00:07:49.820 --> 00:07:51.630
Or this angle plus this
angle plus this

00:07:51.630 --> 00:07:52.560
angle is equal to 180.

00:07:52.560 --> 00:07:53.510
I don't know what I just
said, I think I

00:07:53.510 --> 00:07:56.450
said something wrong.

00:07:56.450 --> 00:07:57.230
Write that down.

00:07:57.230 --> 00:08:00.220
So this angle is going
to be equal to 180

00:08:00.220 --> 00:08:02.780
minus 52 minus 48.

00:08:02.780 --> 00:08:06.460
Because the sum of the angles
add up to 180, and so that is

00:08:06.460 --> 00:08:16.870
equal to 180 minus 100 which
equals 80 degrees.

00:08:16.870 --> 00:08:20.530
So this angle right here
is equal to 80 degrees.

00:08:20.530 --> 00:08:22.830
And the angle they want us to
figure out is the opposite of

00:08:22.830 --> 00:08:25.200
this angle, or in the
U.S., I guess, they

00:08:25.200 --> 00:08:27.630
say vertical angles.

00:08:27.630 --> 00:08:30.770
And so opposite or vertical
angles are equal or they're

00:08:30.770 --> 00:08:35.080
congruent, so this is going
to be 80 degrees as well.

00:08:35.080 --> 00:08:36.750
And that is choice A.

00:08:39.309 --> 00:08:44.390
Problem 50: What is the measure
of an exterior angle

00:08:44.390 --> 00:08:47.000
of a regular hexagon?

00:08:47.000 --> 00:08:49.560
A regular hexagon tells us that
all of the sides are the

00:08:49.560 --> 00:08:51.750
same, it's equilateral, and
all of the angles are the

00:08:51.750 --> 00:08:53.870
same, equiangular.

00:08:53.870 --> 00:08:57.350
So if we just knew what's the
total degree measure of the

00:08:57.350 --> 00:09:01.630
interior angles, we could just
divide that by 6, and then

00:09:01.630 --> 00:09:04.710
that would give us what each
of the interior angles are,

00:09:04.710 --> 00:09:06.210
and then we could use that
information to figure out the

00:09:06.210 --> 00:09:06.890
exterior angles.

00:09:06.890 --> 00:09:07.500
Let's just do it.

00:09:07.500 --> 00:09:11.180
So once again, I like to
just draw a hexagon.

00:09:11.180 --> 00:09:14.480
Let's just draw a hexagon and
count the triangles in it.

00:09:14.480 --> 00:09:18.440
Two sides, three sides,
four sides, five

00:09:18.440 --> 00:09:21.070
sides and six sides.

00:09:21.070 --> 00:09:23.300
And how many triangles
do I have here?

00:09:23.300 --> 00:09:27.650
One, two, three.

00:09:27.650 --> 00:09:33.710
So I have one, two, three,
four triangles.

00:09:33.710 --> 00:09:36.400
The sum of the interior angles
of this hexagon, of any

00:09:36.400 --> 00:09:39.150
hexagon, whether it's regular
or not, are going to be 4

00:09:39.150 --> 00:09:51.810
times 180 and that's
720 degrees.

00:09:51.810 --> 00:09:54.640
And it's a regular hexagon, so
all the interior angles are

00:09:54.640 --> 00:09:55.390
going to be the same.

00:09:55.390 --> 00:09:56.600
And there's six of them.

00:09:56.600 --> 00:10:03.690
So each of them are going
to be 720 divided by 6.

00:10:03.690 --> 00:10:09.030
Well, 6 goes into
72 twelve times.

00:10:09.030 --> 00:10:15.250
So each of the interior angles
are going to be 120 degrees.

00:10:15.250 --> 00:10:17.160
And I didn't draw it that
regular, but we can assume

00:10:17.160 --> 00:10:21.640
that all of these are
each 120 degrees.

00:10:21.640 --> 00:10:22.790
Fair enough.

00:10:22.790 --> 00:10:25.220
Now, if all of those are each
120 degrees, what is the

00:10:25.220 --> 00:10:26.790
measure of an exterior angle?

00:10:26.790 --> 00:10:29.150
Well, we could just extend
one of these sides

00:10:29.150 --> 00:10:31.440
out a little bit.

00:10:31.440 --> 00:10:35.260
We could say, OK, if this is
120 degrees, what is its

00:10:35.260 --> 00:10:36.670
supplement?

00:10:36.670 --> 00:10:38.760
Well, these have to add
up to 180, so 180

00:10:38.760 --> 00:10:41.560
minus 120 is 60 degrees.

00:10:41.560 --> 00:10:42.410
I could do it on any side.

00:10:42.410 --> 00:10:45.390
I could extend that line out
there, and I'd say, oh, that's

00:10:45.390 --> 00:10:46.300
60 degrees.

00:10:46.300 --> 00:10:49.680
So any of the exterior angles
are 60 degrees.

00:10:49.680 --> 00:10:51.030
B.

00:10:51.030 --> 00:10:54.180
All right, do I have
time for one more?

00:10:54.180 --> 00:10:55.665
I'll wait for the next one
in the next video.

00:10:55.665 --> 00:10:57.190
See you soon.


CA Geometry: Exterior angles

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:02.760
We're on 41.

00:00:02.760 --> 00:00:06.890
Lea made two candles in the
shape of right rectangular

00:00:06.890 --> 00:00:10.810
prisms. So I'm assuming when
they say right rectangular

00:00:10.810 --> 00:00:12.670
prisms, they mean
a kind of three

00:00:12.670 --> 00:00:14.570
dimensional rectangular shape.

00:00:14.570 --> 00:00:17.370
The first candle is 15
centimeters height, 8

00:00:17.370 --> 00:00:20.300
centimeters long and
8 centimeters wide.

00:00:20.300 --> 00:00:22.990
So let's see, it's 15
centimeters high.

00:00:22.990 --> 00:00:25.090
So that's 15.

00:00:25.090 --> 00:00:26.640
8 centimeters long.

00:00:26.640 --> 00:00:28.420
So maybe that's 8.

00:00:28.420 --> 00:00:31.030
And 8 centimeters wide.

00:00:31.030 --> 00:00:34.740
So maybe it goes back 8.

00:00:34.740 --> 00:00:36.710
So it looks something
like that.

00:00:36.710 --> 00:00:38.030
That's candle number one.

00:00:38.030 --> 00:00:40.540
The second candle is 5
centimeters higher, but the

00:00:40.540 --> 00:00:41.950
same length and width.

00:00:41.950 --> 00:00:45.400
So the second candle is just
5 centimeters higher.

00:00:45.400 --> 00:00:49.900
So it looks something
like this.

00:00:49.900 --> 00:00:51.410
Where this is still 8 and 8.

00:00:51.410 --> 00:00:55.220
But the height is 5 more
than 15, so it's 20.

00:00:55.220 --> 00:00:57.030
Fair enough.

00:00:57.030 --> 00:00:59.040
How much additional
wax was needed to

00:00:59.040 --> 00:01:00.590
make the taller candle?

00:01:00.590 --> 00:01:02.280
So if you think about it, we
just have to think about how

00:01:02.280 --> 00:01:06.430
much incremental volume did we
create by making that section

00:01:06.430 --> 00:01:07.990
five centimeters higher?

00:01:07.990 --> 00:01:09.620
So this candle, you can
kind of view it

00:01:09.620 --> 00:01:11.430
as going up to here.

00:01:11.430 --> 00:01:14.560
It's 15 centimeters high.

00:01:14.560 --> 00:01:16.610
And then we added
5 right here.

00:01:16.610 --> 00:01:21.980
So what's the volume of this
volume right there?

00:01:21.980 --> 00:01:24.195
So it's 8 by 8 by 5.

00:01:28.660 --> 00:01:31.900
So 5 times 8 times 8.

00:01:31.900 --> 00:01:36.670
5 times 8 is 40 times
8 is 320.

00:01:36.670 --> 00:01:41.450
So we have to add 320 cubic
centimeters more of wax to

00:01:41.450 --> 00:01:44.460
make the taller candle.

00:01:44.460 --> 00:01:46.900
Problem 42.

00:01:46.900 --> 00:01:51.450
Two angles of a triangle have
measures of 55 and 65.

00:01:51.450 --> 00:01:54.220
Which of the following could
not be a measure of an

00:01:54.220 --> 00:01:57.160
exterior angle of
the triangle?

00:01:57.160 --> 00:01:59.740
So I think this is a good time
to introduce what an exterior

00:01:59.740 --> 00:02:01.270
angle even is.

00:02:01.270 --> 00:02:04.390
So if I draw any polygon, and
I'll draw a triangle since

00:02:04.390 --> 00:02:05.983
that's what this question
is about.

00:02:09.449 --> 00:02:12.060
So let's say that that's
my triangle.

00:02:12.060 --> 00:02:15.520
An exterior angle of one
of the vertices is, you

00:02:15.520 --> 00:02:19.820
essentially extend one of the
lines of the vertices out.

00:02:19.820 --> 00:02:22.140
So this is an interior
angle right here.

00:02:22.140 --> 00:02:24.640
The exterior angle is if you
extend this line out, so if I

00:02:24.640 --> 00:02:26.406
were to draw a dotted
line that extends

00:02:26.406 --> 00:02:27.650
out this bottom line.

00:02:27.650 --> 00:02:30.770
This is the exterior
angle right here.

00:02:30.770 --> 00:02:33.510
As you can see, it's going to
be the supplement to this

00:02:33.510 --> 00:02:34.800
interior angle.

00:02:34.800 --> 00:02:37.040
And we could have extended
the line out there.

00:02:37.040 --> 00:02:41.170
Or we could have extended
this line this way.

00:02:41.170 --> 00:02:42.580
And we could have
used this one.

00:02:42.580 --> 00:02:44.590
But we wouldn't add these two
if we wanted to find all of

00:02:44.590 --> 00:02:45.415
the exterior angles.

00:02:45.415 --> 00:02:48.490
The exterior angle of this
vertex right here is either

00:02:48.490 --> 00:02:49.640
this one or this one.

00:02:49.640 --> 00:02:52.960
And they are the same because
both of these are supplements

00:02:52.960 --> 00:02:53.910
of this angle.

00:02:53.910 --> 00:02:57.740
This angle plus either of this
angle or that one will add up

00:02:57.740 --> 00:02:59.230
to 180 degrees.

00:02:59.230 --> 00:03:01.590
So that's what an exterior
angle is.

00:03:01.590 --> 00:03:03.290
So let's go back to
the question.

00:03:03.290 --> 00:03:06.190
Two angles of a triangle have
measures of 55 and 65.

00:03:06.190 --> 00:03:11.960
So let's say this is
55 and this is 65.

00:03:11.960 --> 00:03:13.820
Which of the following could
not be a measure of an

00:03:13.820 --> 00:03:15.430
exterior angle of
the triangle?

00:03:15.430 --> 00:03:17.920
Well we can figure out all
of the exterior angles.

00:03:17.920 --> 00:03:20.720
So first of all what's
this third interior

00:03:20.720 --> 00:03:22.220
angle going to be?

00:03:22.220 --> 00:03:23.600
Well they all have
to add up to 180.

00:03:23.600 --> 00:03:26.160
So let's call that x.

00:03:26.160 --> 00:03:35.790
So we know that x plus 65
plus 55 is equal to 180.

00:03:35.790 --> 00:03:38.870
65 plus 55 is 120.

00:03:38.870 --> 00:03:40.700
120 is equal to 180.

00:03:40.700 --> 00:03:43.170
So x is equal to 60 degrees.

00:03:43.170 --> 00:03:46.340
So this angle right here, I'll
do it in another color, this

00:03:46.340 --> 00:03:47.710
is 60 degrees.

00:03:47.710 --> 00:03:49.430
So what are all the possible
exterior angles.

00:03:49.430 --> 00:03:52.370
So if I extended this line out
like I did in the example of

00:03:52.370 --> 00:03:55.070
when I defined what an exterior
angle is, this

00:03:55.070 --> 00:03:57.840
exterior angle would
be 120 degrees.

00:03:57.840 --> 00:04:00.480
If were to do it here, if I
would extend this out right

00:04:00.480 --> 00:04:04.250
here, what would this
exterior angle be?

00:04:04.250 --> 00:04:07.130
Let's see, this plus
65 is 180.

00:04:07.130 --> 00:04:08.890
What's 180 minus 65.

00:04:08.890 --> 00:04:12.870
180 minus 60 is 120, so this
would have to be 115.

00:04:12.870 --> 00:04:14.970
So that exterior angle is 115.

00:04:14.970 --> 00:04:18.990
And then this one, let's
see if I extend it out.

00:04:18.990 --> 00:04:22.700
One of the two lines that
form the vertex.

00:04:22.700 --> 00:04:25.160
This is going to be
supplementary to 55.

00:04:25.160 --> 00:04:29.760
So 180 minus 55 is 125.

00:04:29.760 --> 00:04:34.010
180 minus 60 would be 120, and
then it's only 55 so 125.

00:04:34.010 --> 00:04:38.120
So the three supplementary, or
the three exterior angles of

00:04:38.120 --> 00:04:40.100
this triangle are 125.

00:04:40.100 --> 00:04:42.110
And they want to know what
could not be a measure.

00:04:42.110 --> 00:04:44.430
So 125 is a measure of
an exterior angle.

00:04:44.430 --> 00:04:45.950
So is 115.

00:04:45.950 --> 00:04:47.400
And so is 120.

00:04:47.400 --> 00:04:48.830
So our answer is D.

00:04:48.830 --> 00:04:54.200
None of the exterior angles
are equal to 130 degrees.

00:04:54.200 --> 00:05:09.320
Problem 43.

00:05:09.320 --> 00:05:13.050
OK, they say the sum of the
interior angles of a polygon

00:05:13.050 --> 00:05:17.060
is the same as the sum of its
exterior angles, what type of

00:05:17.060 --> 00:05:19.130
polygon is it?

00:05:19.130 --> 00:05:22.650
And this is an interesting
question.

00:05:22.650 --> 00:05:24.810
And it's something to experiment
with for yourself.

00:05:24.810 --> 00:05:28.340
But I want you to draw random
polygons with angle measures,

00:05:28.340 --> 00:05:30.140
because you know what the angles
all have to add up to

00:05:30.140 --> 00:05:31.230
in a polyogon.

00:05:31.230 --> 00:05:34.500
And I think you'll find, that
no matter what polygon you

00:05:34.500 --> 00:05:37.540
draw, all of the exterior angles
are going to add up to

00:05:37.540 --> 00:05:39.360
360 degrees.

00:05:39.360 --> 00:05:41.120
In fact, in that example we just
did, what were they they

00:05:41.120 --> 00:05:42.780
were, for that triangle.

00:05:42.780 --> 00:05:49.510
If I remember, it's
115, 125, and 120.

00:05:49.510 --> 00:05:50.370
This was for a triangle.

00:05:50.370 --> 00:05:53.800
If you added them up, you
get 5 plus 5, 10.

00:05:53.800 --> 00:05:55.380
And then that's 6.

00:05:55.380 --> 00:05:56.670
360 degrees.

00:05:56.670 --> 00:06:00.110
For that triangle, which had
kind of strange angles.

00:06:00.110 --> 00:06:00.890
It wasn't like an equilateral

00:06:00.890 --> 00:06:02.440
triangle or anything beautiful.

00:06:02.440 --> 00:06:06.300
And it's also the same if I
were to draw a rectangle.

00:06:06.300 --> 00:06:09.075
Well let me not draw
a solid rectangle.

00:06:11.750 --> 00:06:14.090
So if I have a rectangle
like that.

00:06:14.090 --> 00:06:17.280
What are the exterior
angles here?

00:06:17.280 --> 00:06:21.380
Well, I can continue this
line right here.

00:06:21.380 --> 00:06:25.210
This angle right here
is going to be 90.

00:06:25.210 --> 00:06:27.640
I could go either way, I could
continue this up, but you can

00:06:27.640 --> 00:06:30.970
only do it once though for
each of the vertices.

00:06:30.970 --> 00:06:33.280
Well that exterior
angle is 90.

00:06:33.280 --> 00:06:36.125
I could go like that, that
exterior angle is 90.

00:06:36.125 --> 00:06:37.050
I could go like that.

00:06:37.050 --> 00:06:38.830
That exterior angle is 90.

00:06:38.830 --> 00:06:42.740
So once again, 90 plus 90 plus
90 plus 90 that's 360 degrees.

00:06:42.740 --> 00:06:45.740
So it's a good thing to know
that the sum of the exterior

00:06:45.740 --> 00:06:49.490
angles of any polygon is
actually 360 degrees.

00:06:49.490 --> 00:06:53.910
And maybe we'll prove that in
another video for a polygon

00:06:53.910 --> 00:06:55.030
with n sides.

00:06:55.030 --> 00:06:57.910
But now that we know that, so
they say that the sum of the

00:06:57.910 --> 00:07:00.930
interior angles of a polygon is
the same as the sum of its

00:07:00.930 --> 00:07:04.000
exterior angles, this is the
same as saying the sum of the

00:07:04.000 --> 00:07:06.060
interior angles is
equal to 360.

00:07:06.060 --> 00:07:09.510
Because this is always going
to be 360 degrees no matter

00:07:09.510 --> 00:07:10.910
what the polygon is.

00:07:10.910 --> 00:07:15.560
So they're essentially saying
what polygon's interior angles

00:07:15.560 --> 00:07:17.180
add up to 360 degrees.

00:07:17.180 --> 00:07:20.390
And that of course is
a quadrilateral.

00:07:20.390 --> 00:07:22.980
My mouth got ahead of me.

00:07:22.980 --> 00:07:26.410
And if you think in a
quadrilateral you have 90, 90,

00:07:26.410 --> 00:07:29.910
90, 90 and they add
up to 360 degrees.

00:07:29.910 --> 00:07:32.270
Next question.

00:07:32.270 --> 00:07:35.080
Let me copy and paste a couple
of them so I don't have to

00:07:35.080 --> 00:07:36.330
keep doing this.

00:07:42.120 --> 00:07:43.940
OK.

00:07:43.940 --> 00:07:45.390
All right.

00:07:45.390 --> 00:07:49.140
What is a measure of angle x.

00:07:49.140 --> 00:07:52.760
So this is an exterior angle
to the vertex B.

00:07:52.760 --> 00:07:54.510
So how do we figure this out?

00:07:54.510 --> 00:07:57.160
Well there's kind of a fast
way and a slow way.

00:07:57.160 --> 00:08:00.690
And the slow way is to figure
out this angle.

00:08:00.690 --> 00:08:03.310
Because you know that the sum
of the angles add up to 180.

00:08:03.310 --> 00:08:12.430
And you say oh, x is going
to be 180 minus that.

00:08:12.430 --> 00:08:14.100
Let's just do it the slow way
and I think you'll see the

00:08:14.100 --> 00:08:16.440
intuition of a slightly faster
way you could have done it.

00:08:16.440 --> 00:08:23.370
This plus 60 plus 25
is 85 degrees.

00:08:23.370 --> 00:08:25.270
Let's call this angle y.

00:08:25.270 --> 00:08:32.020
So we know that y plus 85
degrees is equal to 180.

00:08:32.020 --> 00:08:34.730
I got this 85 just by
adding 60 to 25.

00:08:34.730 --> 00:08:38.840
So this is just saying that
the interior angles of a

00:08:38.840 --> 00:08:44.080
triangle add up to
180 degrees.

00:08:44.080 --> 00:08:45.380
And we could figure
out y right now.

00:08:45.380 --> 00:08:48.990
You could subtract 85 from both
sides and you'd get y is

00:08:48.990 --> 00:08:53.100
equal to 95.

00:08:53.100 --> 00:08:56.480
And then we could figure out
x from y, because x is the

00:08:56.480 --> 00:08:57.400
supplement of y.

00:08:57.400 --> 00:09:02.090
So then you could say x is
equal to 180 minus 95 and

00:09:02.090 --> 00:09:03.850
you'd get 85.

00:09:03.850 --> 00:09:06.185
And that would be fine, it
didn't take you too long, C is

00:09:06.185 --> 00:09:06.840
the answer.

00:09:06.840 --> 00:09:08.710
But a slightly faster
way of saying it.

00:09:08.710 --> 00:09:11.460
OK, y plus 85 is equal to 180.

00:09:11.460 --> 00:09:16.930
And you also know that y
plus x is equal to 180.

00:09:16.930 --> 00:09:20.490
So clearly, x is equal to 85.

00:09:20.490 --> 00:09:23.600
If you add 85 to y, you
get 180, if you add x

00:09:23.600 --> 00:09:24.720
to y you get 180.

00:09:24.720 --> 00:09:25.630
So x would be 85.

00:09:25.630 --> 00:09:26.930
That would be a slightly
faster way of

00:09:26.930 --> 00:09:27.650
thinking about it.

00:09:27.650 --> 00:09:29.260
But either way is
fine if you're

00:09:29.260 --> 00:09:31.220
not under time pressure.

00:09:31.220 --> 00:09:34.010
OK, problem 45.

00:09:34.010 --> 00:09:37.960
If the measure of an exterior
angle of a regular polygon.

00:09:37.960 --> 00:09:40.770
Regular polygon, so that
means that all of

00:09:40.770 --> 00:09:43.950
the angles are congruent.

00:09:43.950 --> 00:09:46.980
Of a regular polygon is 120
degrees, how many sides does

00:09:46.980 --> 00:09:48.230
the polygon have?

00:09:52.700 --> 00:09:57.320
So this is the vertex in
question, let's say that's the

00:09:57.320 --> 00:09:59.870
vertex of this polygon
we're thinking about.

00:09:59.870 --> 00:10:01.510
We want to measure its
exterior angles.

00:10:01.510 --> 00:10:04.240
So I'd extend one side
of the vertex.

00:10:04.240 --> 00:10:08.190
And they're saying that
that is 120 degrees.

00:10:08.190 --> 00:10:11.370
That tells me that the interior
angle at that vertex

00:10:11.370 --> 00:10:13.250
is 60 degrees.

00:10:13.250 --> 00:10:15.910
It's the supplement to
the exterior angle.

00:10:15.910 --> 00:10:18.860
So what regular polygon
has all of its

00:10:18.860 --> 00:10:21.970
sides equal to 60 degrees?

00:10:21.970 --> 00:10:25.170
Well, the equilateral
triangle.

00:10:25.170 --> 00:10:26.890
Regular polygon, all
the angles and all

00:10:26.890 --> 00:10:28.410
the sides are congruent.

00:10:28.410 --> 00:10:33.360
So an equilateral triangle
looking something like that

00:10:33.360 --> 00:10:36.470
would do the trick.

00:10:36.470 --> 00:10:39.200
It's a regular polygon, all
the sides are the same.

00:10:39.200 --> 00:10:44.780
And its angles are
60, 60 and 60.

00:10:44.780 --> 00:10:47.020
So when they say how many sides
does the polygon have?

00:10:47.020 --> 00:10:49.510
It has three, it's a triangle.

00:10:49.510 --> 00:10:50.600
I'm out of time.

00:10:50.600 --> 00:10:52.740
I'll see you in the
next video.


CA Geometry: Area, circumference, volume

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.950
We're on problem 31.

00:00:02.950 --> 00:00:06.280
A sewing club is making a
quilt consisting of 25

00:00:06.280 --> 00:00:09.650
squares, with each side of
the square measuring 30

00:00:09.650 --> 00:00:10.240
centimeters.

00:00:10.240 --> 00:00:13.000
OK, and there's going
to be 25 of these.

00:00:13.000 --> 00:00:15.260
And they're going
to be 30 by 30.

00:00:15.260 --> 00:00:20.580
If the quilt has 5 rows and 5
columns, what is the perimeter

00:00:20.580 --> 00:00:21.560
of the quilt?

00:00:21.560 --> 00:00:24.190
OK, so let me draw that out.

00:00:24.190 --> 00:00:27.865
So it has 5 rows and 5 columns
and they're all squares.

00:00:27.865 --> 00:00:31.005
So the quilt itself is
going to be a square.

00:00:31.005 --> 00:00:39.050
It has 4 rows, one two, three,
four, and then five.

00:00:39.050 --> 00:00:45.490
And then one, two,
three, four.

00:00:45.490 --> 00:00:47.230
So that's a 5 by 5.

00:00:47.230 --> 00:00:51.970
And each of these squares,
their sides are 30

00:00:51.970 --> 00:00:53.030
centimeters.

00:00:53.030 --> 00:00:56.120
So how long is one side of
this quilt going to be?

00:00:56.120 --> 00:00:58.280
It's going to be 30 times 5.

00:00:58.280 --> 00:01:00.800
So it's going to be
150 centimeters.

00:01:00.800 --> 00:01:03.446
Same argument, that's going
to be 150 centimeters.

00:01:03.446 --> 00:01:06.410
And that's going to be
150 centimeters.

00:01:06.410 --> 00:01:08.230
30 times 5.

00:01:08.230 --> 00:01:10.520
And this is going to
be 150 centimeters.

00:01:10.520 --> 00:01:14.260
So the perimeter is 150 plus
150 plus 150 plus 150 and

00:01:14.260 --> 00:01:16.280
that's 600.

00:01:16.280 --> 00:01:19.590
So that's choice C.

00:01:19.590 --> 00:01:21.330
32.

00:01:21.330 --> 00:01:24.370
The four sides of this figure
will be folded up and taped to

00:01:24.370 --> 00:01:25.520
make a box.

00:01:25.520 --> 00:01:26.530
Fair enough.

00:01:26.530 --> 00:01:28.010
What will be the volume
of the box?

00:01:28.010 --> 00:01:32.790
OK, so if we cut this out right
now and we folded it

00:01:32.790 --> 00:01:36.100
along where I'm drawing these
green lines, we'll get a box.

00:01:36.100 --> 00:01:38.870
And they want to say,
what's the volume?

00:01:38.870 --> 00:01:40.720
Well, volume is just
the base times the

00:01:40.720 --> 00:01:42.200
height times the depth.

00:01:42.200 --> 00:01:44.150
So if I were to fold up this
box is going to look

00:01:44.150 --> 00:01:45.580
something like this.

00:01:45.580 --> 00:01:50.210
You're going to have the base,
which is this base right here.

00:01:50.210 --> 00:01:52.970
And this is one, one, two,
three, four, five by one, two,

00:01:52.970 --> 00:01:54.370
three, four, five.

00:01:54.370 --> 00:01:56.660
So it's 5 by 5 base.

00:01:56.660 --> 00:02:00.470
And then each of the sides are
going to be 2 high if I fold

00:02:00.470 --> 00:02:02.440
this up, it's going to
be 2 high like that.

00:02:06.060 --> 00:02:07.910
It's going to look like that
if I folded that side up.

00:02:07.910 --> 00:02:12.140
This side, when I fold it up,
is going to look like this.

00:02:12.140 --> 00:02:14.340
One, two, three, four, five.

00:02:14.340 --> 00:02:18.860
That's side when I fold it is
going to look like that.

00:02:18.860 --> 00:02:21.700
And this side when I fold it up
is going to look like that.

00:02:21.700 --> 00:02:24.230
One, two, three, four, five.

00:02:24.230 --> 00:02:27.250
The big picture, the width is
5, the depth is 5, and the

00:02:27.250 --> 00:02:28.360
height is 2.

00:02:28.360 --> 00:02:32.330
So the volume is 5 times
5 is 35 times 2.

00:02:32.330 --> 00:02:35.470
Which is equal to 50.

00:02:35.470 --> 00:02:38.230
And that's choice A.

00:02:38.230 --> 00:02:41.070
Problem 33.

00:02:41.070 --> 00:02:43.650
Where's 33, I think it's
on the next page.

00:02:46.400 --> 00:02:48.210
OK, let me copy and paste it.

00:02:53.330 --> 00:02:57.120
I should just copy and
paste the whole test.

00:02:57.120 --> 00:03:01.020
OK, it says a classroom globe
has a diameter of 18 inches.

00:03:01.020 --> 00:03:03.080
If I were to go from
the center to the

00:03:03.080 --> 00:03:05.700
side, it's 18 inches.

00:03:05.700 --> 00:03:09.090
Which of the following is the
approximate surface area.

00:03:09.090 --> 00:03:10.590
Oh no, sorry, I just
drew the radius.

00:03:10.590 --> 00:03:12.280
It has a diameter
of 18 inches.

00:03:12.280 --> 00:03:15.490
This is 18.

00:03:15.490 --> 00:03:17.760
Which of the following is an
approximate surface area in

00:03:17.760 --> 00:03:19.480
square inches of the globe?

00:03:19.480 --> 00:03:21.520
And surface area, they give us
the equation, they give it in

00:03:21.520 --> 00:03:23.650
terms of the radius.

00:03:23.650 --> 00:03:26.210
So if the diameter is 18,
what's the radius?

00:03:26.210 --> 00:03:27.460
The radius is half
of the diameter.

00:03:30.090 --> 00:03:34.050
So the radius is equal to 9.

00:03:34.050 --> 00:03:35.580
And we just plug
that into here.

00:03:35.580 --> 00:03:39.140
So the surface area is equal
to 4 pi times the radius

00:03:39.140 --> 00:03:41.420
squared, times 9 squared.

00:03:41.420 --> 00:03:47.110
That equals 4 times
81 times pi.

00:03:47.110 --> 00:03:51.010
So it's 324 pi.

00:03:51.010 --> 00:03:52.350
And they actually multiplied
it out.

00:03:52.350 --> 00:03:53.810
So let's see.

00:03:53.810 --> 00:04:01.620
Let's see, 324 pi.

00:04:01.620 --> 00:04:06.320
And if I were to guess this, I
mean, look at all the choices.

00:04:06.320 --> 00:04:09.470
Pi is more than 3.

00:04:09.470 --> 00:04:12.820
So this value is going to be
more than 3 times 324.

00:04:12.820 --> 00:04:16.230
So it's going to be around 1,000
or a little bit more

00:04:16.230 --> 00:04:17.160
than 1,000.

00:04:17.160 --> 00:04:20.240
And the only one that's even
close to that is D.

00:04:20.240 --> 00:04:25.070
But if you wanted to confirm
that, you could multiply 324

00:04:25.070 --> 00:04:35.010
times 3.14 and that is
equal to 1,017.4.

00:04:35.010 --> 00:04:37.030
All right, next problem.

00:04:37.030 --> 00:04:37.830
Problem 34.

00:04:37.830 --> 00:04:52.480
I'll copy and paste 34 and
35 at the same time.

00:04:52.480 --> 00:04:54.270
There you go.

00:04:54.270 --> 00:04:55.520
I put this there.

00:04:58.230 --> 00:05:00.720
All right, ready to do it.

00:05:00.720 --> 00:05:03.750
The rectangle shown below has
a length of 20 meters and

00:05:03.750 --> 00:05:04.460
width of 10.

00:05:04.460 --> 00:05:09.380
So this is 10 and this is 20.

00:05:09.380 --> 00:05:12.100
I just picked that because this
looks longer than that.

00:05:12.100 --> 00:05:13.600
That's a 10 I drew.

00:05:13.600 --> 00:05:15.200
I know it doesn't
look like a 10.

00:05:15.200 --> 00:05:18.490
If the four triangles are
removed from the rectangle as

00:05:18.490 --> 00:05:21.000
shown, what will be the area
of the remaining figure?

00:05:21.000 --> 00:05:22.320
So what's the are before
I remove them?

00:05:22.320 --> 00:05:23.980
It's 20 times 10.

00:05:23.980 --> 00:05:25.910
That's the area of the
whole rectangle.

00:05:25.910 --> 00:05:27.390
So it's 200.

00:05:27.390 --> 00:05:29.270
And then how much area
am I removing?

00:05:29.270 --> 00:05:32.130
So each of these triangles,
what's its area?

00:05:32.130 --> 00:05:34.810
It's base times height
times 1/2.

00:05:34.810 --> 00:05:36.920
That's the area of a triangle.

00:05:36.920 --> 00:05:38.400
Because if you just did base
times height, you'd be

00:05:38.400 --> 00:05:42.040
figuring out the area of this
little rectangle there.

00:05:42.040 --> 00:05:47.200
So the area of this is 4 times 4
is 16 times 1/2, which is 8.

00:05:47.200 --> 00:05:50.400
This is going to be 8, going
to be 8, going to be 8.

00:05:50.400 --> 00:05:52.720
So we're removing four
8's from this area.

00:05:52.720 --> 00:05:54.520
So we're removing 32.

00:05:54.520 --> 00:05:56.410
So minus 32.

00:05:56.410 --> 00:05:59.710
And that's 168.

00:05:59.710 --> 00:06:01.095
So that's choice C.

00:06:04.430 --> 00:06:05.880
Problem 35.

00:06:05.880 --> 00:06:14.780
If RSTW is a rhombus, so rhombus
tells us that all the

00:06:14.780 --> 00:06:16.960
sides are equal and
they're parallel.

00:06:16.960 --> 00:06:19.450
What is the area of WXT.

00:06:26.440 --> 00:06:27.740
So this right here.

00:06:33.440 --> 00:06:37.250
OK, so this is something that
you may or may not have

00:06:37.250 --> 00:06:38.960
learned about a rhombus
already.

00:06:38.960 --> 00:06:43.580
But its diagonals actually
intersect at a

00:06:43.580 --> 00:06:46.210
perpendicular line.

00:06:46.210 --> 00:06:49.740
And let me see what
else we can see.

00:06:49.740 --> 00:06:54.020
This is 60 degrees, so
this is 30 degrees.

00:06:54.020 --> 00:06:55.990
Let's see what we can
get from this.

00:06:55.990 --> 00:06:58.300
This is 12.

00:06:58.300 --> 00:07:01.540
Then this is 12.

00:07:01.540 --> 00:07:03.680
OK, I see where they're
going with this.

00:07:03.680 --> 00:07:07.740
So if this is 90 degrees,
this is 90 degrees.

00:07:07.740 --> 00:07:12.280
This is a rhombus, so all
the sides are the same.

00:07:12.280 --> 00:07:18.370
If this is 60, this is 90, this
has to be 30 degrees.

00:07:18.370 --> 00:07:21.320
And then you could actually make
a very strong argument

00:07:21.320 --> 00:07:22.650
that these are similar
triangles.

00:07:22.650 --> 00:07:25.300
Whatever length this is,
that's the same length,

00:07:25.300 --> 00:07:27.860
because this is a parallelogram
and the

00:07:27.860 --> 00:07:29.350
diagonals bisect each other.

00:07:29.350 --> 00:07:30.710
This side is equal
to that side.

00:07:30.710 --> 00:07:31.730
That side is equal
to this side.

00:07:31.730 --> 00:07:34.220
So these are congruent
triangles.

00:07:34.220 --> 00:07:37.450
So this is also going
to be 60 degrees.

00:07:37.450 --> 00:07:39.590
This is going to be 30.

00:07:39.590 --> 00:07:43.400
But if you have a 60, let me do
it in another color, if you

00:07:43.400 --> 00:07:47.720
have a 60, 60, 60 triangle,
all of the angles are 60

00:07:47.720 --> 00:07:50.560
degrees, you're dealing with
an equilateral triangle.

00:07:50.560 --> 00:07:55.960
So that tells you that all
the sides are the same.

00:07:55.960 --> 00:07:58.580
So if this side is 12, that side
is 12, this side right

00:07:58.580 --> 00:08:04.650
here also has to be 12.

00:08:04.650 --> 00:08:07.280
If that whole side is 12,
what's this length?

00:08:07.280 --> 00:08:10.320
We already know that in a
parallelogram the diagonals

00:08:10.320 --> 00:08:11.220
bisect each other.

00:08:11.220 --> 00:08:12.820
So this length is 6.

00:08:12.820 --> 00:08:15.330
And this length is 6.

00:08:15.330 --> 00:08:16.750
Fair enough.

00:08:16.750 --> 00:08:20.430
And let's see, if each of these
lengths are 6, can we

00:08:20.430 --> 00:08:23.850
figure out what this
height is equal to?

00:08:23.850 --> 00:08:26.790
Because if we know the base and
the height, we're ready to

00:08:26.790 --> 00:08:29.390
figure out the area
of a triangle.

00:08:29.390 --> 00:08:30.820
So let's see if we can use
the Pythagorean Theorem.

00:08:30.820 --> 00:08:35.950
If we called this x, we could
say x squared plus 6 squared

00:08:35.950 --> 00:08:41.490
plus 36 is equal to 12 squared,
is equal to 144.

00:08:41.490 --> 00:08:45.580
And we you could say that x
squared is equal to, what's

00:08:45.580 --> 00:08:52.200
144 minus 36, that's 108.

00:08:52.200 --> 00:08:55.660
Let's see, 108.

00:08:55.660 --> 00:08:58.882
x squared is equal to 108.

00:08:58.882 --> 00:09:01.750
x is equal to the square
root of 108.

00:09:01.750 --> 00:09:04.430
And I can simplify that
more because 9

00:09:04.430 --> 00:09:10.680
goes into 108 12 times.

00:09:10.680 --> 00:09:12.210
Let me do that.

00:09:12.210 --> 00:09:14.240
So x is equal to the
square root of 9

00:09:14.240 --> 00:09:16.250
times 12, that's 108.

00:09:16.250 --> 00:09:18.900
So that's equal to the square
root of 9 times the square

00:09:18.900 --> 00:09:23.380
root of 12.

00:09:23.380 --> 00:09:28.190
That's equal to 3 times
the square root of 12.

00:09:28.190 --> 00:09:30.030
Square root of 12 is the same
thing as the square root of 3

00:09:30.030 --> 00:09:32.640
times the square root of 4.

00:09:32.640 --> 00:09:33.720
Square root of 4 is 2.

00:09:33.720 --> 00:09:37.830
So that's 2 times 3 is
6 square root of 3.

00:09:37.830 --> 00:09:39.260
This is 36 times 3.

00:09:39.260 --> 00:09:41.870
We could have said this is equal
to the square root of 36

00:09:41.870 --> 00:09:43.210
times the square root of 3.

00:09:47.260 --> 00:09:49.360
All right, so 6 square
roots of 3.

00:09:49.360 --> 00:09:51.510
That's this side.

00:09:51.510 --> 00:09:53.940
So what's the area of this
triangle right there?

00:09:53.940 --> 00:09:59.280
It's 1/2 times this base times
6 times 6 square roots of 3.

00:10:01.800 --> 00:10:05.040
So that's 1/2 times 6 is 3,
times, 6 square roots of 3 is

00:10:05.040 --> 00:10:07.170
18 square roots of 3.

00:10:07.170 --> 00:10:09.280
Now that's just this triangle.

00:10:09.280 --> 00:10:11.970
This triangle is congruent to
this triangle, so it will have

00:10:11.970 --> 00:10:12.450
the same area.

00:10:12.450 --> 00:10:14.100
And you can make the same
argument that all of these

00:10:14.100 --> 00:10:15.440
triangles are congruent.

00:10:15.440 --> 00:10:17.176
So the area of this entire
rhombus is going

00:10:17.176 --> 00:10:18.530
to be 4 times this.

00:10:23.520 --> 00:10:24.540
Is that what they wanted?

00:10:24.540 --> 00:10:26.930
Oh no, they wanted of
the area of WXD.

00:10:26.930 --> 00:10:28.000
That's what we just
figured out.

00:10:28.000 --> 00:10:29.250
They didn't want the area
of the whole rhombus.

00:10:29.250 --> 00:10:31.880
They want just the area of this
triangle right there,

00:10:31.880 --> 00:10:37.170
which we just figured out, which
is 18 square roots of 3.

00:10:37.170 --> 00:10:40.060
I'm trying to think if there's
a simpler way of doing this.

00:10:40.060 --> 00:10:42.510
There might be some formula for
the equation for the area

00:10:42.510 --> 00:10:44.790
of a rhombus that I've forgotten
in my memory.

00:10:44.790 --> 00:10:46.120
But we were able
to re-prove it.

00:10:46.120 --> 00:10:49.770
And that's always better, to
come from basic principles.

00:10:49.770 --> 00:10:52.220
Anyway, I'll see you
in the next video.


CA Geometry: Pythagorean theorem, area

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.510
We're on problem 36.

00:00:02.510 --> 00:00:06.560
And it says, what is the area
in square units of the

00:00:06.560 --> 00:00:09.180
trapezoid shown below?

00:00:09.180 --> 00:00:11.930
So, when you just look at this
you're like, OK, trapezoid, do

00:00:11.930 --> 00:00:14.580
I know the formula for the
area of a trapezoid.

00:00:14.580 --> 00:00:15.990
Then you might get confused
and all that.

00:00:15.990 --> 00:00:18.520
But you say, well, trapezoid
I can break that up into a

00:00:18.520 --> 00:00:20.120
rectangle and a triangle.

00:00:20.120 --> 00:00:22.935
If were to draw a
line right here.

00:00:26.110 --> 00:00:28.220
Then I've broken up the
trapezoid into a rectangle and

00:00:28.220 --> 00:00:29.030
a triangle.

00:00:29.030 --> 00:00:32.720
And if I know the dimensions of
each of those, I know the

00:00:32.720 --> 00:00:34.480
area of each of them and
then I know the area

00:00:34.480 --> 00:00:35.980
of the entire thing.

00:00:35.980 --> 00:00:39.790
So let's see, what's this height
right Here Or this

00:00:39.790 --> 00:00:43.000
width I should say.

00:00:43.000 --> 00:00:45.430
Well we're going from
zero to what?

00:00:45.430 --> 00:00:48.480
x is equal to 8 here.

00:00:48.480 --> 00:00:50.390
I just went straight down
from x is equal to 8,

00:00:50.390 --> 00:00:51.260
y is equal to 5.

00:00:51.260 --> 00:00:52.720
So this dimension is 8.

00:00:52.720 --> 00:00:56.190
And then we go from x is equal
to 8 to x is equal to 12, how

00:00:56.190 --> 00:00:57.210
far is that?

00:00:57.210 --> 00:00:59.060
Well, that's going to be 4.

00:00:59.060 --> 00:01:02.840
So this is 4 and this is 8.

00:01:02.840 --> 00:01:04.080
Fair enough.

00:01:04.080 --> 00:01:06.000
And then how high is
this rectangle?

00:01:06.000 --> 00:01:08.380
We're going from y is equal
to zero to y is equal to

00:01:08.380 --> 00:01:10.030
5, so that's 5.

00:01:10.030 --> 00:01:12.035
And of course this
is 5 as well.

00:01:12.035 --> 00:01:13.450
So we're done.

00:01:13.450 --> 00:01:14.620
We're ready to figure
out the area.

00:01:14.620 --> 00:01:18.730
The area of the rectangle part
is 8 times 5, that's 40.

00:01:18.730 --> 00:01:23.030
The area of this triangle
is 5 times 4 times 1/2.

00:01:23.030 --> 00:01:24.690
If we didn't put that 1/2 we
would be figuring out the area

00:01:24.690 --> 00:01:26.310
of this rectangle right there.

00:01:26.310 --> 00:01:29.710
So 5 times 4 is 20
times 1/2 is 10.

00:01:29.710 --> 00:01:36.050
So the area of both of these
combined is 10 plus 40 is 50.

00:01:36.050 --> 00:01:37.770
37.

00:01:37.770 --> 00:01:40.380
The figure below is a square
with 4 congruent

00:01:40.380 --> 00:01:41.615
parallelograms in side.

00:01:41.615 --> 00:01:43.030
This looks interesting.

00:01:43.030 --> 00:01:46.180
What is the area in square units
of the shaded portion?

00:01:46.180 --> 00:01:50.240
So the shaded portion is the
whole square minus the area of

00:01:50.240 --> 00:01:50.960
the parallelogram.

00:01:50.960 --> 00:01:52.780
So the whole square, that's
easy, it's 12.

00:01:52.780 --> 00:01:55.540
And the height is 12, but since
you know it's a square

00:01:55.540 --> 00:01:58.830
we know the width also
has to be 12.

00:01:58.830 --> 00:02:04.670
So the area of the whole
square is 144.

00:02:04.670 --> 00:02:06.700
If we know the area of one of
the parallelograms, we know

00:02:06.700 --> 00:02:08.930
the area of all of the
parallelograms because they

00:02:08.930 --> 00:02:10.419
are congruent.

00:02:10.419 --> 00:02:12.140
So let's see if we can figure
out the area of one of the

00:02:12.140 --> 00:02:13.870
parallelograms.

00:02:13.870 --> 00:02:16.840
So there is actually a formula
for the area of a

00:02:16.840 --> 00:02:18.200
parallelogram, it's
actually just the

00:02:18.200 --> 00:02:20.360
base times the height.

00:02:20.360 --> 00:02:21.420
And they actually
give us that.

00:02:21.420 --> 00:02:23.600
But let me show you that they
give us that because it might

00:02:23.600 --> 00:02:26.090
not be obvious to you.

00:02:26.090 --> 00:02:28.510
Let me try to draw it.

00:02:28.510 --> 00:02:31.370
I'll use my line tool.

00:02:31.370 --> 00:02:32.900
Nope, that's not
the line tool.

00:02:36.260 --> 00:02:41.430
One side, then go straight
like that, come down like

00:02:41.430 --> 00:02:45.590
that, good enough.

00:02:45.590 --> 00:02:49.000
OK, now if I look at just this
parallelgoram, they tell us

00:02:49.000 --> 00:02:50.320
that the height here is 3.

00:02:53.280 --> 00:02:54.650
And I know it's the height
because they told me it's a 90

00:02:54.650 --> 00:02:55.400
degree angle.

00:02:55.400 --> 00:02:58.930
And they tell us at
the base is 5.

00:02:58.930 --> 00:03:01.250
And I'm telling you that the
area of a parallelogram is

00:03:01.250 --> 00:03:04.340
just the base times the
height is equal to 15.

00:03:04.340 --> 00:03:06.010
But you shouldn't just
take my word for it.

00:03:06.010 --> 00:03:08.000
That should make intuitive
sense to you.

00:03:08.000 --> 00:03:10.970
And the way to think about it
intuitively is imagine if we

00:03:10.970 --> 00:03:17.970
were to take this part of the
parallelogram, and if we were

00:03:17.970 --> 00:03:19.750
to move it over here.

00:03:19.750 --> 00:03:22.710
If we were to cut that off
and move it over here.

00:03:22.710 --> 00:03:25.460
Then the parallelogram would
look something like this.

00:03:25.460 --> 00:03:29.250
You'd have the part that
we didn't cut off.

00:03:29.250 --> 00:03:31.250
And then you move the cut-off
part over here.

00:03:33.940 --> 00:03:37.230
And now the dimensions, this
base would be 5, and this

00:03:37.230 --> 00:03:38.110
height would be 3.

00:03:38.110 --> 00:03:41.040
And the area of this
rectangle is 15.

00:03:41.040 --> 00:03:42.850
And there's no reason why the
area of this should be any

00:03:42.850 --> 00:03:43.460
different than that.

00:03:43.460 --> 00:03:45.240
We just rearranged its parts.

00:03:45.240 --> 00:03:48.130
So that's why the area of a
parallelogram is just the base

00:03:48.130 --> 00:03:49.160
times the height.

00:03:49.160 --> 00:03:52.700
So the area of each of these
parallelograms is 15.

00:03:52.700 --> 00:03:55.050
So the area of all of
them combined is 15

00:03:55.050 --> 00:03:56.960
times 4, which is 60.

00:03:56.960 --> 00:04:03.655
So 144 minus 60 is 84.

00:04:03.655 --> 00:04:08.250
And that's choice B.

00:04:08.250 --> 00:04:09.680
Problem 38.

00:04:09.680 --> 00:04:11.690
What is the area in square
meters of the

00:04:11.690 --> 00:04:14.000
trapezoid shown below.

00:04:14.000 --> 00:04:16.510
So to figure out the area and
we could break it up into

00:04:16.510 --> 00:04:17.970
these rectangles
and triangles.

00:04:17.970 --> 00:04:19.700
To figure out the area
of this rectangle we

00:04:19.700 --> 00:04:22.300
need to know its height.

00:04:22.300 --> 00:04:26.170
And actually we'll need that to
figure out the area of the

00:04:26.170 --> 00:04:27.670
triangles as well.

00:04:27.670 --> 00:04:30.250
So what's this height
right there?

00:04:30.250 --> 00:04:34.030
Let's see, we know that this
distance is going to be 6.

00:04:34.030 --> 00:04:35.280
It's a rectangle.

00:04:37.430 --> 00:04:41.550
If that distance is 6 and both
of these are 5, both of these

00:04:41.550 --> 00:04:44.920
triangles here are going
to be congruent.

00:04:44.920 --> 00:04:48.090
Because this length is
equal to this length.

00:04:48.090 --> 00:04:51.600
This length is equal
to this length.

00:04:51.600 --> 00:04:56.590
And we also we make this angle
is equal to that angle.

00:04:56.590 --> 00:05:01.200
But anyway, let me do
it in another color.

00:05:01.200 --> 00:05:04.370
What's the length of these
two green sides?

00:05:04.370 --> 00:05:07.180
Let's call it x.

00:05:07.180 --> 00:05:09.460
Well we know that when you
add x plus 6 plus x it

00:05:09.460 --> 00:05:10.440
has to equal 12.

00:05:10.440 --> 00:05:12.070
The whole top part.

00:05:12.070 --> 00:05:16.850
So you get x plus x is 2x
plus 6 is equal to 12.

00:05:16.850 --> 00:05:18.410
2x is equal to 6.

00:05:18.410 --> 00:05:19.840
x is equal to 3.

00:05:19.840 --> 00:05:21.410
And you might have been able
to solve that in your head.

00:05:21.410 --> 00:05:23.990
That if that's 6 and these are
the same, then both of these

00:05:23.990 --> 00:05:26.630
are going to be 3.

00:05:26.630 --> 00:05:28.870
And now we can use that
information to figure out this

00:05:28.870 --> 00:05:31.170
height right there.

00:05:31.170 --> 00:05:36.230
Because if we just draw this
triangle right there, that's

00:05:36.230 --> 00:05:40.970
3, that's 5, this is some
unknown side, a.

00:05:40.970 --> 00:05:42.690
You might already recognize,
we're going to use the

00:05:42.690 --> 00:05:43.390
Pythagorean theorem.

00:05:43.390 --> 00:05:46.490
And this is a very typical
type of right triangle.

00:05:46.490 --> 00:05:48.390
So might already be able
to guess what a is.

00:05:48.390 --> 00:05:49.680
But we'll solve for it.

00:05:49.680 --> 00:05:54.150
So we know that a squared plus
3 squared is equal to the

00:05:54.150 --> 00:05:57.510
hypotenuse squared, the side
opposite the 90 degree angle.

00:05:57.510 --> 00:06:00.230
So that's equal to 25.

00:06:00.230 --> 00:06:02.230
5 squared is 25.

00:06:02.230 --> 00:06:06.020
a squared plus 9
is equal to 25.

00:06:06.020 --> 00:06:08.190
a squared is equal to 16.

00:06:08.190 --> 00:06:10.420
a is equal to 4.

00:06:10.420 --> 00:06:11.580
a is equal to 4.

00:06:11.580 --> 00:06:14.260
And now we're ready to
figure out the area.

00:06:14.260 --> 00:06:15.920
What's the area of
the rectangle?

00:06:15.920 --> 00:06:18.140
6 times 6, it's 24.

00:06:18.140 --> 00:06:20.590
What's the area of each
of these triangles?

00:06:20.590 --> 00:06:23.730
3 times 4 times 1/2.

00:06:23.730 --> 00:06:27.020
3 times 4 is 12 times
1/2 is 6.

00:06:27.020 --> 00:06:28.620
So the area of that
triangle is 6.

00:06:28.620 --> 00:06:30.310
The area of this
triangle is 6.

00:06:30.310 --> 00:06:35.050
So 6 plus 24 plus 6 is 36.

00:06:35.050 --> 00:06:36.930
B.

00:06:36.930 --> 00:06:40.010
Problem 39.

00:06:40.010 --> 00:06:44.660
What is the area in square
inches of the triangle below.

00:06:44.660 --> 00:06:45.800
Interesting.

00:06:45.800 --> 00:06:48.710
OK, so this is an equilateral
triangle, all

00:06:48.710 --> 00:06:50.070
the sides are equal.

00:06:50.070 --> 00:06:53.270
And so we could actually say
that since these two triangles

00:06:53.270 --> 00:06:54.790
are symmetric.

00:06:54.790 --> 00:06:56.160
That's equal to that.

00:06:56.160 --> 00:06:58.820
And this comes to a general
formula for the area of an

00:06:58.820 --> 00:06:59.330
equilateral triangle.

00:06:59.330 --> 00:07:00.770
But let's just figure
it all out.

00:07:00.770 --> 00:07:03.740
So this side is going to be 5.

00:07:03.740 --> 00:07:06.390
And this side is
going to be 5.

00:07:06.390 --> 00:07:09.810
If this is 5 and that's 10, what
is this side right here?

00:07:09.810 --> 00:07:11.520
Let's call it x.

00:07:11.520 --> 00:07:13.600
Pythagorean theorem.

00:07:13.600 --> 00:07:14.520
This is the hypotenuse.

00:07:14.520 --> 00:07:20.080
So x squared plus 5 squared plus
25 is going to be equal

00:07:20.080 --> 00:07:23.440
to the hypotenuse squared,
it's equal to 100.

00:07:23.440 --> 00:07:27.460
x squared is equal to
100 minus 25, 75.

00:07:27.460 --> 00:07:30.860
x is equal to the square
root of 75.

00:07:30.860 --> 00:07:32.900
75 is 25 times 3.

00:07:32.900 --> 00:07:36.710
So that's equal to the square
root of 25 times 3.

00:07:36.710 --> 00:07:39.390
Which is equal to the square
root of 25 times the

00:07:39.390 --> 00:07:41.830
square root of 3.

00:07:41.830 --> 00:07:44.250
Which is equal to
5 roots of 3.

00:07:48.960 --> 00:07:51.410
And now, what's the area
of just this right

00:07:51.410 --> 00:07:52.380
triangle right here?

00:07:52.380 --> 00:07:54.080
This one on the right side.

00:07:54.080 --> 00:07:57.000
Well its base is 5, its height
is 5 roots of 3.

00:07:57.000 --> 00:08:03.310
So it's going to be 1/2 times
the base, 5, times the height,

00:08:03.310 --> 00:08:06.080
5 roots of 3.

00:08:06.080 --> 00:08:07.360
And that's what?

00:08:07.360 --> 00:08:08.730
1/2 times 5 times 5.

00:08:08.730 --> 00:08:14.380
So it's 25 root 3 over 2 and
that's just this triangle

00:08:14.380 --> 00:08:15.460
right there.

00:08:15.460 --> 00:08:19.720
Well this triangle's going to
have the the exact same area.

00:08:19.720 --> 00:08:21.540
They are congruent triangles.

00:08:21.540 --> 00:08:25.530
So the area of the figure
is this times 2.

00:08:25.530 --> 00:08:29.410
So 2 times that is equal
to just the 25 root 3.

00:08:29.410 --> 00:08:33.000
And that's choice B.

00:08:33.000 --> 00:08:35.700
Next problem, problem 40.

00:08:35.700 --> 00:08:39.600
The perimeter of two squares
are in a ratio of 4:9.

00:08:39.600 --> 00:08:42.250
What is the ratio between the
areas of the two squares?

00:08:42.250 --> 00:08:45.630
Let me draw two squares.

00:08:45.630 --> 00:08:47.300
That's one square.

00:08:47.300 --> 00:08:49.090
Let me draw another square.

00:08:49.090 --> 00:08:51.610
That's another square.

00:08:51.610 --> 00:08:58.210
Let's say that the sides of this
are x and the sides of

00:08:58.210 --> 00:08:59.460
this one are y.

00:09:01.900 --> 00:09:04.460
So they say the perimeters of
those two squares are in a

00:09:04.460 --> 00:09:05.460
ratio of 4:9.

00:09:05.460 --> 00:09:07.890
So the perimeter of the
first square is 4x.

00:09:07.890 --> 00:09:10.190
x plus x plus x plus x.

00:09:10.190 --> 00:09:12.770
So the perimeter of the
first square is 4x.

00:09:12.770 --> 00:09:16.540
The perimeter of the second
square is 4y.

00:09:16.540 --> 00:09:18.880
So that's the ratio of the
perimeter of the first square

00:09:18.880 --> 00:09:21.400
to the perimeter of
the second square.

00:09:21.400 --> 00:09:23.600
And then that is equal to 4:9.

00:09:28.420 --> 00:09:30.800
And they say, what is the ratio
between the areas of the

00:09:30.800 --> 00:09:31.450
two squares?

00:09:31.450 --> 00:09:33.685
So they want us to figure out
the area of the first square

00:09:33.685 --> 00:09:36.700
is x squared.

00:09:36.700 --> 00:09:38.630
Base times height, x times x.

00:09:38.630 --> 00:09:42.580
And the are of the second
square is y times y.

00:09:42.580 --> 00:09:47.700
So they want us to figure out
what that is equal to.

00:09:47.700 --> 00:09:50.140
Well this is x squared
over y squared.

00:09:50.140 --> 00:09:55.690
This is the same thing
as x over y squared.

00:09:55.690 --> 00:09:58.150
So if we can figure out what x
over y is equal to, we can

00:09:58.150 --> 00:10:00.430
just square it and we'll get
x squared over y squared.

00:10:00.430 --> 00:10:02.290
So let's try to do that.

00:10:02.290 --> 00:10:03.540
So they gave us this.

00:10:06.130 --> 00:10:09.235
Well this just simplifies.

00:10:09.235 --> 00:10:11.213
x over y is equal to 4 over 9.

00:10:14.780 --> 00:10:16.210
So let's substitute that here.

00:10:16.210 --> 00:10:20.360
So x squared over y squared is
equal to x over y squared.

00:10:20.360 --> 00:10:23.910
Which is equal to 4/9 squared.

00:10:23.910 --> 00:10:28.430
Which is equal to 16 over 81.

00:10:28.430 --> 00:10:32.680
Or the ratio of the areas of
the two squares is 16:81.

00:10:32.680 --> 00:10:35.830
Choice D.

00:10:35.830 --> 00:10:39.060
I think we can fit one more
problem in there.

00:10:39.060 --> 00:10:40.330
Actually no, I'm over
10 minutes.

00:10:40.330 --> 00:10:41.510
I'll stop right there.

00:10:41.510 --> 00:10:43.360
See you in the next video.