2003 AIME II problem 12

WEBVTT
Kind: captions
Language: en

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The members of a
distinguished committee

00:00:03.140 --> 00:00:05.790
were choosing a
president, and each member

00:00:05.790 --> 00:00:09.355
gave one vote to one
of the 27 candidates.

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So it looks like they all
got at least one vote.

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For each candidate,
the exact percentage

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of votes the candidate got
was smaller by at least one

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than the number of votes
for that candidate.

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Let me underline this
again and read it again.

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It's kind of a
confusing statement.

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For each candidate,
the exact percentage

00:00:34.140 --> 00:00:39.000
of votes the candidate got
was smaller by at least one

00:00:39.000 --> 00:00:41.520
than the number of votes
for that candidate.

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What is the smallest
possible number

00:00:51.070 --> 00:00:54.006
of members of the committee?

00:00:54.006 --> 00:00:56.005
So let's define some
variables here and maybe we

00:00:56.005 --> 00:00:57.860
can get our heads around this.

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So let's just say that m is
equal to the number of members

00:01:01.497 --> 00:01:02.205
of the committee.

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That's equal to the
number of members.

00:01:07.780 --> 00:01:09.530
And let's just think
about each candidate.

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Let's say that we
have 27 candidates,

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but let's just pick some
arbitrary candidate i,

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so that i could be any number
from one to 27, any integer.

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This sentence right here,
let me write in blue

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since I underlined it in blue.

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Let me write it here.

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Candidate c sub i is
equal to the number

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of votes for the ith candidate.

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Now let's see if
we can do something

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with what I underlined in blue.

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The exact percentage of
votes the candidate got

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was smaller-- OK.

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So the exact percentage
of votes the candidate got

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would be the votes he
got or she got divided

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by the total number of members.

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Now this will give
you a decimal.

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If you want it as a
percentage, you're

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going to want to
multiply it by 100.

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So times 100.

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So that's this part.

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The exact percentage of
votes the candidate got,

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so that's this right here,
was smaller by at least one

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than the number of votes
for that candidate.

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So smaller by at least one.

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So the number of votes of
the candidate is c sub i,

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and it's not less than or
equal to the number of votes

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they got.

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It's less than or equal to 1
less than the number of votes

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they got, at least 1 less.

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So we're going to
put a minus 1 here.

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So this is telling us that
the percentage of votes

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that any candidate got is
less than the number of votes

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they got minus 1,
which is really

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what that sentence is saying.

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And frankly, that's the hardest
part about this problem,

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is understanding what that
sentence is even saying.

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Now what is the
smallest possible number

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of members of the committee?

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Now think about it.

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We want to minimize
the number of members.

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So we really want to
minimize the number of votes

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that each candidate gets.

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So let's just think about
the minimum number of votes

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for each candidate,
per candidate.

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So can we have 0
votes for a candidate?

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Well, they tell us here.

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They say each
member gave one vote

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to one of the 27 candidates.

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So everyone got
at least one vote.

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So no.

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We cannot have 0 votes.

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Can we have one
vote per candidate?

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It seems fair that they
each got at least one vote.

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But if we look over
here, their percentage

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having to be 1 less than the
number of votes they have,

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and that's what we
wrote over here,

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that makes 1 pretty difficult.

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Let's think about
this a little bit.

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Let's think about it.

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If the minimum
number of votes is

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1, that means for
that candidate who

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got the minimum number of
votes, the right-hand side

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of this equation right
here, this right-hand side

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of the equation
would be-- sorry.

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My cell phone is ringing.

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My apologies.

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So where I left off,
we were thinking,

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can we have one
vote for a candidate

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or can there be a candidate
that only has one vote?

00:04:06.060 --> 00:04:08.051
Can the minimum votes
per candidate be one?

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And we were looking
at the right-hand side

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of this equation we set
up from the word problem.

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And if it was 1, this
right here would be 0.

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And this, on the left hand
side, if it's at least 1--

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we know that we have a
positive number of members.

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And we know that this is
going to be at least 1.

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So this is going to be
some positive fraction

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that we're going
to multiply by 100.

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So there's no way that that
can be less than or equal to 0.

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So even the guy or the gal
who got the fewest votes

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has to get more than one vote.

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So this also can't be the case.

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This expression right
over here can never

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be less than or equal to
0, which would be the case

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if the minimum number
of votes was one.

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So let's think
about more of them.

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Let's see if we can
ever have two votes

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for a candidate, a
minimum of two votes.

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And to do that, what I want to
do-- if we put 2 in here-- what

00:05:02.990 --> 00:05:05.720
I want to do is I want to
see what repercussions that

00:05:05.720 --> 00:05:07.331
has for our members.

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Does it put any threshold
for our members?

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And to do that,
let me solve for m.

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It actually simplifies
a little bit now

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that we've put the constraint
that our minimum number

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of votes can never
be equal to 1.

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So what I want to
do is, let's take

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the inverse of both
sides of this equation.

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Let me do this in a new color.

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Let me do it in pink.

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Let's take the inverse of
both sides of this equation.

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The left hand side over here, we
essentially have 100 ci over m,

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so that will be m
over 100 c sub i.

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And since we're inverting
both sides of the equation,

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this less than or equal is
going to become a greater than

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or equal to.

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And then we're going to
have 1 over c sub i minus 1.

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So I just inverted both sides.

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You swap the inequality.

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Obviously, if you had 1 is less
than 2, now if you invert them,

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1 is greater than 1/2.

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So that's the logic behind
swapping the inequality.

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And now, if I want
to solve for m,

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I can multiply both
sides by 100ci,

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and that's a
positive value, so it

00:06:05.390 --> 00:06:07.390
won't do anything
to the inequality.

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So m is going to be greater than
or equal to 100 c i, the number

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of votes the ith candidate
got over ci minus 1.

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So let's see.

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Can a candidate get 2 votes?

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So if a candidate got 2
votes-- if the candidate

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with the fewest votes
got 2, what repercussions

00:06:25.520 --> 00:06:29.150
would that have based on
what we just solved for?

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Well, in that
situation, m would have

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to be greater than or equal
to 100 times 2 over 2 minus 1.

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So m would have to
be greater than 200.

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Seems reasonable.

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Maybe our answer is 200, or
greater than or equal to 200.

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So maybe our answer's 200.

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But let's try it with some
larger minimum number of votes.

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Maybe that can bring
down our m a little bit.

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So what if we had 3?

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What does this
inequality tell us?

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Then m would have to be greater
than or equal to 300 divided

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by 3 minus 1, 300 divided by 2.

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Yep.

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It did come down.

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So it looks like as we increase
our minimum number of votes

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per candidate, we're
getting this threshold for m

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to come down.

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So let's try higher numbers.

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If our minimum vote per
candidate is 4, then we have m

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is greater than or
equal to 400 over 3.

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So m would have to be greater
than or equal to 133 1/3.

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And obviously, this has to
be an integer right here.

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So the smallest possible m in
this situation would be 134.

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Well, it seems like we're
making progress here.

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Let's try 5.

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If we have a minimum
number of votes of 5,

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then m has to be greater than
or equal to 500 divided by 4

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is 125.

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Now this looks
tempting, but remember,

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this is the minimum number
of votes per candidate,

00:08:00.050 --> 00:08:02.010
and we have 27 candidates.

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So what does this tell
us about the minimum--

00:08:06.409 --> 00:08:08.200
if this is the minimum
votes per candidate,

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what is the minimum
number of votes,

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not even thinking about
this threshold here.

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Well, it's going to be
this number times 27.

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In this case, if you had
two votes for candidate

00:08:19.140 --> 00:08:21.980
you would have to have 54.

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That's 27 times 2.

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27 times 3 is 81.

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27 times 4 is what.

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That's 80, that's 108.

00:08:33.195 --> 00:08:38.919
27 times 5 is 135.

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So even though having
at least 5 votes

00:08:41.400 --> 00:08:44.720
per candidate kind of
loosens your constraint on m,

00:08:44.720 --> 00:08:46.410
if you have 5 votes
per candidate,

00:08:46.410 --> 00:08:48.780
you're going to have 135 votes.

00:08:48.780 --> 00:08:50.820
You can't get down to 125 votes.

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If you have 4 votes
per candidate--

00:08:52.970 --> 00:08:55.690
in order to have 4 times
27, you only need 108.

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So if you bump that
108 to 134-- Well,

00:09:00.950 --> 00:09:02.770
there's two ways you
could get 134 votes.

00:09:02.770 --> 00:09:06.540
You can give 26 people
5 votes, and then

00:09:06.540 --> 00:09:08.860
give the last person
4 votes, and that

00:09:08.860 --> 00:09:10.360
would meet the
constraint over here.

00:09:10.360 --> 00:09:12.180
It gets you 1 less than 135.

00:09:12.180 --> 00:09:14.450
Or you could give
26 people 4 votes

00:09:14.450 --> 00:09:16.000
and give the last
person-- I think

00:09:16.000 --> 00:09:19.090
it would come out
to be 30 votes.

00:09:19.090 --> 00:09:21.590
And either way, this would work.

00:09:21.590 --> 00:09:24.817
So our answer is if
we have 134 votes--

00:09:24.817 --> 00:09:27.400
or I guess another way to say
it is our answer to the question

00:09:27.400 --> 00:09:30.040
is the minimum number
of committee members

00:09:30.040 --> 00:09:32.980
we need is 134.

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Hopefully, you found
that interesting.


2003 AIME II problem 13

WEBVTT
Kind: captions
Language: en

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A bug starts at a vertex of an equilateral traingle. On each move, it randomly selects one of the two

00:00:07.563 --> 00:00:14.453
vertices where it is not currently located and crawls along a side of the triangle to that vertex.

00:00:14.453 --> 00:00:21.419
Given that the probability that the bug moves to its starting vertex on its tenth move is m/n

00:00:21.419 --> 00:00:25.501
where m and n are relatively prime positive integers,

00:00:25.501 --> 00:00:26.652
find m + n

00:00:26.652 --> 00:00:30.074
So this lesson right here where m and n are relatively prime positive integers

00:00:30.074 --> 00:00:35.670
essentially says we have the positive version of this fraction in its simplified form

00:00:35.670 --> 00:00:42.677
So let's just think about the problem we have an equilateral triangle here

00:00:42.677 --> 00:00:46.545
with three vertices

00:00:46.545 --> 00:00:49.413
A B and C

00:00:49.413 --> 00:00:54.376
and our bug is going to start at lets say vertex A

00:00:54.376 --> 00:00:57.745
So this is our bug right here

00:00:57.745 --> 00:01:01.227
and on each move it will randomly select one fo the two vertices

00:01:01.227 --> 00:01:07.260
So on its first move it will either go to vertex B or then vertex C

00:01:07.260 --> 00:01:09.077
and then depending on whether it went there

00:01:09.077 --> 00:01:12.162
If it went to vertex C then on it's next move it wil either go to vertex B or vertex A

00:01:12.162 --> 00:01:18.877
if it went to vertex B on its next move it will either go to vertex A or vertex C

00:01:18.877 --> 00:01:22.118
So let's just think about it.

00:01:22.118 --> 00:01:35.004
We want to figure out the probability of it moving to vertex a on its tenth move

00:01:35.004 --> 00:01:38.324
So let's just think about what happened in each move.

00:01:38.324 --> 00:01:42.469
The probability of moving toward on of the vertices in a given move

00:01:42.469 --> 00:01:51.204
So let us say that we have move 1 over here in this column

00:01:51.204 --> 00:01:57.885
so we have vertex A,B and C

00:01:57.885 --> 00:01:59.669
so let's think about move one

00:01:59.669 --> 00:02:09.604
On move 1 what's the probablilty of moving to vertex A

00:02:09.604 --> 00:02:14.204
Well you are already on vertex A and it says that the insect in going to go to one of the other 2 vertices

00:02:14.204 --> 00:02:19.787
So the probability on move one of moving to vertex A is 0

00:02:19.787 --> 00:02:21.720
What's the probablilty of moving to vertex B?

00:02:21.720 --> 00:02:23.021
Well it is going to be 1/2

00:02:23.021 --> 00:02:27.036
what's the probablilty of moving to vertex C?

00:02:27.036 --> 00:02:29.203
It's going to be 1/2 as well

00:02:29.203 --> 00:02:31.036
half a chance of going there and half a chance fo going there

00:02:31.036 --> 00:02:35.704
Well obviously all of the probabilities have to add up to 1 because it's going to do something on that

00:02:35.704 --> 00:02:38.605
move. Now let us think about move 2

00:02:38.605 --> 00:02:48.134
So what is the probability of now moving to vertex A?

00:02:48.134 --> 00:02:57.682
Well there is a 1/2 chance that we are already at B. If we were already at B there will now be a 1/2

00:02:57.682 --> 00:02:59.983
chance of moving back to A

00:02:59.983 --> 00:03:04.462
So this is 1/2 times 1/2

00:03:04.462 --> 00:03:09.767
that's only the situation if we were already at B. If we were already at C

00:03:09.767 --> 00:03:14.937
then there would also be a 1/2 probability of moving to A

00:03:14.937 --> 00:03:24.003
So this is going be the 1/2 chance that we were already at C times the probability that we move to A

00:03:24.003 --> 00:03:32.133
So 1/2 times 1/2 = 1/4 + 1/2 times 1/2 = 1/2

00:03:32.610 --> 00:03:42.417
So what is the probability of moving to B in the second turn? Now the only place other than B

00:03:42.417 --> 00:03:49.651
that the bug can move to is C in the second turn.

00:03:49.651 --> 00:03:54.402
This is because we know that at the end of the first turn the bug will not be at A

00:03:54.402 --> 00:03:56.402
It will either be at B or C

00:03:56.402 --> 00:03:58.668
If the bug is already at B there is no way that it will go back to B

00:03:58.668 --> 00:04:03.829
So the only situation is if the bug is already at C there is a 1/2 chance of that happening

00:04:03.829 --> 00:04:05.317
so that is this number right here.

00:04:05.317 --> 00:04:10.233
So if the bug is already at C and that is the 1/2 probablility then there is going to be a further 1/2

00:04:10.233 --> 00:04:12.184
probability that it moves to B

00:04:12.184 --> 00:04:17.834
So it is going to be 1/2 times 1/2 = 1/4

00:04:17.834 --> 00:04:21.168
Now what is the probability of moving to C? Well it is the same logic.

00:04:21.168 --> 00:04:30.034
The bug is not going to be on vertex A after the first move there is a 1/2 probability that it is going

00:04:30.034 --> 00:04:37.519
to be at B; and then if it is at B then there is a 1/2 probability that it will go to C

00:04:37.519 --> 00:04:43.267
So this is going to be 1/4 as well

00:04:43.267 --> 00:04:58.548
In general let's say that in move N the probablility of being at vertex A is

00:04:58.548 --> 00:05:00.281
the probability of A

00:05:00.281 --> 00:05:06.116
The probability of being at vertex B is probability of landing on B at move N

00:05:06.116 --> 00:05:12.032
and the probability of C is Probability of moving ot vertex C on move N

00:05:12.032 --> 00:05:16.331
Now let us think of the probability of being at each of these nodes on N+1

00:05:16.331 --> 00:05:29.031
So in order to get to A on move n+1, you either have to be at B or C. You cannot be at A and stay at

00:05:29.031 --> 00:05:29.631
A

00:05:29.631 --> 00:05:39.234
So the probability of moving to A in your n+1th move is going to be the probability of being at B times

00:05:39.234 --> 00:05:44.626
1/2 (right because if you are at B there is a one half chance that you are going to go to A) + the probability

00:05:44.626 --> 00:05:51.926
of being at C at the end of your last move times 1/2

00:05:51.926 --> 00:05:56.835
There is a half chance that if you are at C you will move to A. There is a half chance that if you are

00:05:56.835 --> 00:06:00.127
at C you will move to A.

00:06:00.127 --> 00:06:01.759
Now what's the probability of going to B?

00:06:01.759 --> 00:06:03.157
Well there are two ways that will get ot you B.

00:06:03.157 --> 00:06:05.376
You can start at A so you are going to start at A

00:06:05.376 --> 00:06:08.193
So the probablilty of being at A times 1/2

00:06:08.193 --> 00:06:12.775
or you could start at C. So it's plus the probability at C times 1/2

00:06:12.775 --> 00:06:16.309
If you were doing this problem in a situation that had time constraints you would not neccesarily have

00:06:16.309 --> 00:06:18.298
to go through this, but I just want ot make it all clear.

00:06:18.298 --> 00:06:22.724
And finally what is the probability of landing in C at you n+1th term

00:06:22.724 --> 00:06:27.979
It's going to be the probability of being at A times 1/2

00:06:27.979 --> 00:06:32.976
+ the probability of being at B times 1/2

00:06:32.976 --> 00:06:37.242
So hopefully this makes the pattern of what is happening little bit clearer

00:06:37.242 --> 00:06:39.609
So no matter what A is going to be the average.

00:06:39.609 --> 00:06:45.126
A in the next move is essentially going to be the average the probabilities of being at B and C

00:06:45.126 --> 00:06:48.692
And you can see that. The average of 1/2 and 1/2 is 1/2

00:06:48.692 --> 00:06:56.525
The probability of being at B in the n+1th move is going to be the average of the probabilities of being

00:06:56.525 --> 00:07:00.258
at A and C in the last move the average of 0 and 1/2 is 1/4

00:07:00.258 --> 00:07:06.074
and the same logic for C. The average for 0 and 1/2 is 1/4

00:07:06.074 --> 00:07:13.176
Now that will help me simplify things a little bit more for our brains

00:07:13.176 --> 00:07:16.525
Let's keep going this way and let me continue the columns

00:07:16.525 --> 00:07:21.275
Rows A,B and C

00:07:21.275 --> 00:07:25.092
and now we are on move 3. I am just continuing it down here

00:07:25.092 --> 00:07:28.926
So what is the probability of the move being A in move 3

00:07:28.926 --> 00:07:32.376
Well we just said it's going to be the average of B and C and B and C is 1/4

00:07:32.376 --> 00:07:35.725
So the average of 1/4 and 1/4 is going to be 1/4

00:07:35.725 --> 00:07:38.491
What is the probability of being at B?

00:07:38.491 --> 00:07:47.242
Well the average of A and C in move two is 1/2 and 1/4

00:07:47.242 --> 00:07:51.775
that's the same thing as the average of 4/8ths and 2/8ths

00:07:51.775 --> 00:07:54.126
that's 3/8ths

00:07:54.126 --> 00:07:57.391
and then we are going to get the exact same value for C. I think 3/8ths

00:07:57.391 --> 00:08:00.175
Another pattern here is that B and C are always going to be the same thing

00:08:00.175 --> 00:08:04.558
and since B and C are always going to be the same thing the average is going to be that value

00:08:04.558 --> 00:08:08.497
and it's going to be the probability of being at A in the next turn

00:08:08.497 --> 00:08:10.873
So let us just keep doing this

00:08:10.873 --> 00:08:14.006
and we can go all the way to ten but maybe we will see some type of pattern here

00:08:14.006 --> 00:08:21.222
So on out fourth move what is the probability of being at A? Which is the average of B and C

00:08:21.222 --> 00:08:24.473
So it is going to be 3/8ths

00:08:24.473 --> 00:08:28.806
So we can just take this value here and then what is the probability of being at B?

00:08:28.806 --> 00:08:34.089
Well it is going to be the average of A and C

00:08:34.089 --> 00:08:44.566
and so let's see this is 4/16 + 6/16 = 10/16

00:08:44.566 --> 00:08:52.773
and then we have to divide it by two which is 5/16

00:08:52.773 --> 00:08:56.155
Did I do that right? Let me just do this on the corner. I don't want to make a careless mistake here.

00:08:56.155 --> 00:09:06.324
1/4 is 4/16 + 6/16 and then we want to divide that by two.

00:09:06.324 --> 00:09:12.755
We are taking the average. So that's 10/16 divided by 2 which is equal to 5/16

00:09:12.755 --> 00:09:17.139
Alright so let me clear theis out so that I do not waste valuable real estate

00:09:17.139 --> 00:09:20.606
Alright. So then this is also going to be 5/16

00:09:20.606 --> 00:09:22.790
because it's going to be the average of 3/8 and 1/4 as well

00:09:22.790 --> 00:09:29.558
so it's going to be 5/16

00:09:29.558 --> 00:09:32.623
Now we are on move 5

00:09:32.623 --> 00:09:37.174
And we could keep going like this all the way to move ten but hopefully we willl see a pattern here

00:09:37.174 --> 00:09:45.768
So move five the probability of moving to A is going to be 5/16 the average of these two

00:09:45.768 --> 00:09:47.063
5/16

00:09:47.063 --> 00:09:50.318
The probability of moving to B (We can already see a pattern) it seems like we went from having

00:09:50.318 --> 00:09:52.341
2,4,8 in the denominator to 16 in the denominator

00:09:52.618 --> 00:09:56.216
I'm guessing that we are going to have to go to 32 in the denominator

00:09:56.216 --> 00:10:08.083
and so we are going to take the average of 10/32 and 12/32 and that is 11/32

00:10:08.083 --> 00:10:11.649
And this is also going to be 11/32

00:10:11.649 --> 00:10:23.130
Let's go to move 6 for A it's going to be 11/32 (it seems like the pattern is holding) this is going

00:10:23.130 --> 00:10:30.345
ot be something over 64 and so this is the average if you multiply this times 4 you get 20

00:10:30.345 --> 00:10:39.948
and then this is 20/64 and this is 22/64 the average is going to be 21

00:10:39.948 --> 00:10:43.864
So let us keep extrapolating this and see if there is a way

00:10:43.864 --> 00:10:48.947
and we could actually just keep doing the math for 7,8,9,10

00:10:48.947 --> 00:10:53.829
Feel free to do that on your own if you are interested and hopefully you will get the right answer.

00:10:53.829 --> 00:10:56.804
But let's see is you can spot a pattern here

00:10:56.804 --> 00:11:01.111
This is also going to be 21/64 move 7 for A is easy. That's just going to be the average of these two

00:11:01.111 --> 00:11:02.426
guys, 21/64

00:11:02.426 --> 00:11:04.400
But let's see if there is a pattern forming

00:11:04.400 --> 00:11:11.478
So it looks like the (and remember the question is we just want to look at the probability of it moving

00:11:11.478 --> 00:11:14.013
to A on the tenth move.)

00:11:14.013 --> 00:11:20.095
and so if you look at A you start off with a denominator

00:11:20.095 --> 00:11:24.346
So this is 2 to the move minus 1. 1/2 to the move minus 1

00:11:24.346 --> 00:11:28.128
It looks like it's always cuz its 1/2 to the 0. This is 1/2 to the 1

00:11:28.128 --> 00:11:31.479
and you have 1/2 squared that's 1 less than three

00:11:31.479 --> 00:11:32.663
1/2 to the 3rd

00:11:32.663 --> 00:11:39.163
So it looks like if you fast forward to the nth move or even better let us fast forward to the tenth

00:11:39.163 --> 00:11:43.278
move. So 7, 8, 9 and 10

00:11:43.278 --> 00:11:47.329
This is not actually a lot of math to fill in. But let us just figure it out

00:11:47.329 --> 00:11:52.110
So 7,8,9, 10, this is going to be 16, 32, 64

00:11:52.110 --> 00:11:56.206
This is going to be 128 if we go with the pattern

00:11:56.206 --> 00:12:05.646
This is going to be 256 and then this row over here is going to be over 512. So we actually already figured

00:12:05.646 --> 00:12:10.747
out the n part. And lets see is we can figure out a pattern for the numerators here.

00:12:10.747 --> 00:12:13.413
We went from 1 to 3 to 5

00:12:13.413 --> 00:12:18.278
So let us see 5 is 3+2 not 1

00:12:18.278 --> 00:12:26.041
Then we go, let's see you have 5 and it looks like you are taking 2times this and adding it to that to

00:12:26.041 --> 00:12:34.279
get that. 2 times 1 plus 3 is five. 2 times 3 plus 5 is 11. 2 times 5 plus 11 is 21.

00:12:34.279 --> 00:12:37.897
So let us just go with that

00:12:37.897 --> 00:12:50.915
So 2 times 11 is 22 plus 21 is 43 and then 2 times 21 is 42 plus 43 is 85

00:12:50.915 --> 00:13:02.798
and then this is going to be 86 plus 85 is let's see 85 times two is 170 so its going to be 171

00:13:02.798 --> 00:13:07.031
So if we believe what we just did we get out answer

00:13:07.031 --> 00:13:18.030
m + n = 171 +512

00:13:18.030 --> 00:13:22.408
Which is let's see we are going to have two plus 1 is 3

00:13:22.408 --> 00:13:31.364
1 plus so this becomes ten and then we get 703. Wait I made a bone headed mistake on the addition

00:13:31.364 --> 00:13:34.272
Let me do that again

00:13:34.272 --> 00:13:40.021
So 1 plus 7 is 8 not 10. I don't know why my brain was thinking that and then 5 plus 1 is 6. So we get

00:13:40.021 --> 00:13:44.355
683 as our answer

00:13:44.355 --> 00:13:52.356
So that is m + n and the probability of moving to vertex A on the tenth turn is 171/512

00:13:52.356 --> 00:13:54.838
and you can verify this for yourself

00:13:54.838 --> 00:14:03.689
Or we could even prove to ourselves is we are interested that each successive term here is really equal

00:14:03.689 --> 00:14:06.439
to the previous term times two times the term before that

00:14:06.439 --> 00:14:11.322
Actually for fun let us just prove it to ourselves

00:14:11.322 --> 00:14:15.123
So let us just say that this is the nth term

00:14:15.123 --> 00:14:24.972
Our probability of being at A is A/2^n

00:14:24.972 --> 00:14:30.090
and what ends up happening is that the probability of being at b and C let us call that B/

00:14:30.090 --> 00:14:37.722
This power is always 1 higher power of 2. 2 ^n plus 1 and this right over here is also going to be the

00:14:37.722 --> 00:14:40.833
same probability 2/n+1

00:14:40.833 --> 00:14:45.623
Then on the nth + 1 term the probability of being at A is just going to be the average of these two guys

00:14:45.623 --> 00:14:49.872
which is going to be B times 2 to the n plus 1

00:14:49.872 --> 00:14:54.138
and the probability of being at, this is B

00:14:54.138 --> 00:14:56.373
this is C, this is A

00:14:56.373 --> 00:15:06.456
The probability of being at B is going ot be the average of 2A/ 2^(n+1) (I just multiplied the numerator

00:15:06.456 --> 00:15:14.671
and the denominator by 2) plus B/2^(n+1)and then all of that over 2

00:15:14.671 --> 00:15:24.488
So we are going to divide the whole thing by 2

00:15:24.488 --> 00:15:29.337
and that's going to be the same value for C. And actaully if we do a little math here this is going

00:15:29.337 --> 00:15:39.988
to be [2A + B]/[2^(n+2)]

00:15:39.988 --> 00:15:46.572
And so on the nth+2 move our value for A is going to be this thing right here

00:15:46.572 --> 00:15:50.339
The average of these two guys which is the exact same thing so it's going to be

00:15:50.339 --> 00:15:54.872
[2A+B] / [2^(n+2)]

00:15:54.872 --> 00:15:58.238
So this verifies the pattern that we just thought about

00:15:58.238 --> 00:16:02.289
Our powers of two are increasing and so if we go two terms forward

00:16:02.289 --> 00:16:07.874
This term is = to 2 times 2 terms before plus the terms before. 2A+B

00:16:07.874 --> 99:59:59.999
So that is the pattern that we used and we can feel pretty good about the answer!


2003 AIME II problem 8

WEBVTT
Kind: captions
Language: en

00:00:00.555 --> 00:00:02.630
- [Voiceover] Find the
eighth term of the sequence

00:00:02.630 --> 00:00:06.922
1440, 1716, 1848,

00:00:06.922 --> 00:00:08.294
and so on and so forth,

00:00:08.294 --> 00:00:10.732
whose terms are formed by multiplying

00:00:10.732 --> 00:00:15.195
the corresponding terms of
two arithmetic sequences.

00:00:15.195 --> 00:00:16.992
Arithmetic sequences,
always having trouble

00:00:16.992 --> 00:00:18.257
saying that properly.

00:00:18.257 --> 00:00:19.982
So, let's think about the two sequences

00:00:19.982 --> 00:00:22.229
and then we'll take the
products of corresponding terms

00:00:22.229 --> 00:00:24.160
to get the terms of this sequence.

00:00:24.160 --> 00:00:29.014
So maybe one of the arithmetic
sequences starts with A,

00:00:29.889 --> 00:00:32.084
starts with A, and then the next one,

00:00:32.084 --> 00:00:33.607
the next term in that sequence

00:00:33.607 --> 00:00:35.564
is going to be some constant plus A,

00:00:35.564 --> 00:00:37.498
that's what an arithmetic sequence is.

00:00:37.498 --> 00:00:39.996
It's just a constant added
to each successive term.

00:00:39.996 --> 00:00:43.656
So, it's going to be A, A plus M,

00:00:43.656 --> 00:00:47.019
and then the next term
would be M plus A plus M,

00:00:47.019 --> 00:00:49.759
or it'd be A plus two M.

00:00:49.759 --> 00:00:52.125
And if we just kept going, what we care

00:00:52.125 --> 00:00:53.753
is all the way to the eighth term

00:00:53.753 --> 00:00:55.265
because we're gonna take the eight term

00:00:55.265 --> 00:00:56.625
of this sequence, multiply
it by the eighth term

00:00:56.625 --> 00:00:58.359
of the other sequence
to get the eighth term

00:00:58.359 --> 00:01:00.026
of this sequence over here.

00:01:00.026 --> 00:01:01.739
So, the second term, we only added one M,

00:01:01.739 --> 00:01:03.831
the third term, we added two M's,

00:01:03.831 --> 00:01:05.994
so the eighth term, we're
gonna add seven M's,

00:01:05.994 --> 00:01:07.893
one less than the term.

00:01:07.893 --> 00:01:09.761
The first term, we added zero M's.

00:01:09.761 --> 00:01:12.356
So, the eighth term here is going to be,

00:01:12.356 --> 00:01:16.933
is going to be A plus, A plus 7m.

00:01:16.933 --> 00:01:20.784
That's the eighth term of
this arithmetic sequence.

00:01:20.784 --> 00:01:22.823
Now, let's do another one for
another arithmetic sequence,

00:01:22.823 --> 00:01:25.825
maybe that starts at the number B.

00:01:25.825 --> 00:01:28.257
Now, the next term is going
to be some constant plus B,

00:01:28.257 --> 00:01:31.363
so let's just call that B plus N.

00:01:31.363 --> 00:01:34.146
Then the third term
will be N plus B plus N.

00:01:34.146 --> 00:01:37.029
So, it'll be B plus 2n.

00:01:37.029 --> 00:01:38.948
And then we go all the
way to the eighth term

00:01:38.948 --> 00:01:43.754
by the same logic, it's going
to be B plus, B plus 7n.

00:01:43.754 --> 00:01:47.686
The difference between
each of these terms is N.

00:01:47.686 --> 00:01:49.601
Now, they tell us that the product

00:01:49.601 --> 00:01:51.954
of the corresponding terms
of these two sequences

00:01:51.954 --> 00:01:53.891
are the terms of this sequence.

00:01:53.891 --> 00:01:55.954
so, A times B, so when
you take the product

00:01:55.954 --> 00:02:00.229
of A and B, we get A times B,

00:02:00.229 --> 00:02:02.582
A times, I'll color code it for here,

00:02:02.582 --> 00:02:07.324
A times B is going to be equal to 1440.

00:02:07.324 --> 00:02:11.553
1400, 1440.

00:02:11.553 --> 00:02:16.179
We also know that A plus
M, we also know that

00:02:16.179 --> 00:02:21.016
A plus M times B plus N,

00:02:21.016 --> 00:02:26.016
times B plus N is going to be equal to,

00:02:26.052 --> 00:02:30.921
is going to be equal to 1716.

00:02:30.921 --> 00:02:33.629
1716.

00:02:33.629 --> 00:02:38.148
We also know, we also know that A plus 2m,

00:02:38.148 --> 00:02:42.954
A plus 2m times B plus 2n,

00:02:42.954 --> 00:02:47.129
times B plus 2n is going to be

00:02:47.129 --> 00:02:50.472
equal to 1848.

00:02:50.472 --> 00:02:55.472
Is going to be equal to 1848.

00:02:56.226 --> 00:02:58.627
And if we want the eighth
term of this sequence,

00:02:58.627 --> 00:03:01.031
we just have to take this
product right over here.

00:03:01.031 --> 00:03:03.357
So, the eighth term is going to be,

00:03:03.357 --> 00:03:08.357
is going to be A plus 7m, A plus 7m

00:03:08.414 --> 00:03:11.356
times B plus 7n.

00:03:11.356 --> 00:03:15.985
Times B plus, B plus 7n.

00:03:15.985 --> 00:03:17.222
Now, before I even do this, it seems like

00:03:17.222 --> 00:03:18.993
we have some good information to at least

00:03:18.993 --> 00:03:22.539
put some constraints around
our A's, B's, M's, and N's.

00:03:22.539 --> 00:03:23.817
Let's just think about the form

00:03:23.817 --> 00:03:25.143
that this is going to take, and then maybe

00:03:25.143 --> 00:03:26.890
we can kind of build up the numbers

00:03:26.890 --> 00:03:29.417
as we solve for this stuff over here.

00:03:29.417 --> 00:03:32.959
So, the eighth term in our
sequence is going to be,

00:03:32.959 --> 00:03:35.224
I'll do this in a new
color, it's going to be,

00:03:35.224 --> 00:03:38.447
let's just multiply
these two binomials out,

00:03:38.447 --> 00:03:40.822
A times B, it's going to be AB,

00:03:40.822 --> 00:03:43.755
that's A times B, plus A times 7n,

00:03:43.755 --> 00:03:48.755
so that's plus 7an, plus 7m times B,

00:03:49.027 --> 00:03:54.027
so plus 7bm, plus 7m times 7n.

00:03:54.892 --> 00:03:59.122
7m times 7n, so that's plus 49,

00:03:59.122 --> 00:04:04.122
plus 49mn, 49, 49mn.

00:04:04.403 --> 00:04:07.151
So, this is what we need
to figure out the value of.

00:04:07.151 --> 00:04:09.952
We need to figure out the
value of this right here.

00:04:09.952 --> 00:04:12.185
Well, we already know
one of the terms here.

00:04:12.185 --> 00:04:16.488
We already know that A
times b is equal to 1440.

00:04:16.488 --> 00:04:19.562
So, we're already making some head way.

00:04:19.562 --> 00:04:24.232
This right here is going
to be equal to 1440.

00:04:24.232 --> 00:04:26.746
Now, let's see if we can figure
out anything for these terms

00:04:26.746 --> 00:04:29.328
over here using the other
information over here.

00:04:29.328 --> 00:04:31.721
So, let's multiply these out.

00:04:31.721 --> 00:04:34.287
So, let me do this once
again in a new color.

00:04:34.287 --> 00:04:36.364
So, if I multiply this
out, I got A times B,

00:04:36.364 --> 00:04:41.292
which is AB plus A times N, which is AN,

00:04:41.292 --> 00:04:45.022
plus M times B, or B times M, which is BM,

00:04:45.022 --> 00:04:48.483
plus M times N, plus M times N.

00:04:48.483 --> 00:04:52.695
Which is going to be equal to 1716.

00:04:52.695 --> 00:04:56.090
Now, we know that AB is 1440.

00:04:56.090 --> 00:04:59.358
We know that this is 1440.

00:04:59.358 --> 00:05:01.826
So let's see, let's subtract
1440 from both sides

00:05:01.826 --> 00:05:04.022
of this equation, I'll
switch colors again.

00:05:04.022 --> 00:05:07.049
So, let's subtract 1440.

00:05:07.049 --> 00:05:08.763
Well, I guess I didn't switch colors.

00:05:08.763 --> 00:05:10.904
>From both sides of this equation,

00:05:10.904 --> 00:05:12.688
these guys obviously cancel out,

00:05:12.688 --> 00:05:17.274
the left hand side becomes AN plus BM

00:05:17.274 --> 00:05:21.828
plus MN is equal to, let's see,

00:05:21.828 --> 00:05:26.828
the thousands cancel out,
we have a 671 minus 40,

00:05:26.921 --> 00:05:30.228
let's see, we have a
six minus zero is six.

00:05:30.228 --> 00:05:32.592
Then we can borrow, this is a six,

00:05:32.592 --> 00:05:35.523
this is 11, 11 minus four is seven,

00:05:35.523 --> 00:05:37.958
six minus four is two,
one minus one is zero.

00:05:37.958 --> 00:05:42.874
So, AN plus BM plus MN is equal to 276.

00:05:42.874 --> 00:05:44.949
So, we got that information using this

00:05:44.949 --> 00:05:48.163
and that, that, essentially, equation.

00:05:48.163 --> 00:05:50.960
Now let's use this equation over here.

00:05:50.960 --> 00:05:54.227
Do this in a new color,
I'm running out of colors.

00:05:54.227 --> 00:05:55.450
I'll do it in green.

00:05:55.450 --> 00:05:59.146
So, we have A times B, which is AB

00:05:59.146 --> 00:06:00.892
plus A times 2n.

00:06:00.892 --> 00:06:02.952
So, it's 2an.

00:06:02.952 --> 00:06:06.882
Plus 2mb times, 2m times B,

00:06:06.882 --> 00:06:09.831
so it's plus 2bm.

00:06:09.831 --> 00:06:13.153
Plus 2m times 2n, so plus four,

00:06:13.153 --> 00:06:16.731
plus 4mn is equal

00:06:16.731 --> 00:06:19.827
to 1848.

00:06:19.827 --> 00:06:24.207
Well, same idea, this AB, this is 1440.

00:06:24.207 --> 00:06:28.262
We can subtract 1440 from
both sides of this equation.

00:06:28.262 --> 00:06:30.558
I'm running out of space
here so I'm just gonna

00:06:30.558 --> 00:06:34.322
subtract 1440 from the
left and the right side.

00:06:34.322 --> 00:06:37.693
Now this equation becomes,
on the left hand side,

00:06:37.693 --> 00:06:39.606
we're just left with this stuff over here.

00:06:39.606 --> 00:06:41.620
So, we have 2an.

00:06:41.620 --> 00:06:43.670
I'm gonna write it over here
so it doesn't look confusing.

00:06:43.670 --> 00:06:47.731
We have 2an plus 2bm

00:06:47.731 --> 00:06:52.525
plus 4mn is equal to, these
obviously cancelled out.

00:06:52.525 --> 00:06:57.414
848 minus 440 is 408.

00:06:57.414 --> 00:07:00.458
Is equal to 408, did I do that right?

00:07:00.458 --> 00:07:04.494
848, yeah, 408, right.

00:07:04.494 --> 00:07:05.554
And now both sides of this equation

00:07:05.554 --> 00:07:08.190
are divisible by two, so let's
divide both sides by two.

00:07:08.190 --> 00:07:10.189
So, we have, if we
divide everything by two,

00:07:10.189 --> 00:07:13.758
you get AN pus BM

00:07:13.758 --> 00:07:18.758
plus 2mn is equal to,

00:07:18.936 --> 00:07:22.405
is equal to 204.

00:07:23.128 --> 00:07:25.261
Now, these look pretty close.

00:07:25.261 --> 00:07:27.554
It's actually, if we view, if we view this

00:07:27.554 --> 00:07:30.359
as kind of one variable,
and this as another,

00:07:30.359 --> 00:07:31.861
seems like we can solve for MN,

00:07:31.861 --> 00:07:34.353
and then we can solve for AN BM.

00:07:34.353 --> 00:07:36.598
And the reason why that looks useful,

00:07:36.598 --> 00:07:38.137
the reason why that would be useful

00:07:38.137 --> 00:07:40.526
is our final answer, we
have some multiple times MN.

00:07:40.526 --> 00:07:42.922
So, if we know what MN is, we
could put that number here,

00:07:42.922 --> 00:07:45.826
and if we know what AN plus BM is,

00:07:45.826 --> 00:07:47.755
we just multiply that
seven, and we're gonna get

00:07:47.755 --> 00:07:50.801
this term right over
here, so let's do that.

00:07:50.801 --> 00:07:53.957
Let's do that, let's solve for MN.

00:07:53.957 --> 00:07:56.429
And to solve for MN, let's
subtract this equation

00:07:56.429 --> 00:07:57.521
from this equation.

00:07:57.521 --> 00:08:00.292
So, let me do it in that same color.

00:08:00.292 --> 00:08:02.005
So, I'm gonna subtract
this from this equation.

00:08:02.005 --> 00:08:03.428
So, multiply it by negative one.

00:08:03.428 --> 00:08:08.159
We have negative AN minus BM

00:08:08.159 --> 00:08:12.285
minus MN is equal to

00:08:12.285 --> 00:08:16.523
negative 276, is equal to negative 276.

00:08:16.523 --> 00:08:20.001
And then when we subtract, we get,

00:08:20.001 --> 00:08:24.283
these cancel out, these cancel
out, and then 2mn minus MN

00:08:24.283 --> 00:08:25.720
is just gonna be MN.

00:08:25.720 --> 00:08:29.920
And then 204 minus 276, that's negative,

00:08:29.920 --> 00:08:31.694
negative 72.

00:08:31.694 --> 00:08:33.488
So, we're able to solve for MN.

00:08:33.488 --> 00:08:34.866
And that's useful for us because

00:08:34.866 --> 00:08:37.092
this expression here is 49mn.

00:08:37.092 --> 00:08:39.964
So, this number right here,
this number right here

00:08:39.964 --> 00:08:44.360
is negative, that number
right there is negative 72.

00:08:44.360 --> 00:08:46.825
Now, what's AN plus BM?

00:08:46.825 --> 00:08:49.558
Well, we know that this
right here is negative 72.

00:08:49.558 --> 00:08:53.289
This right here is negative 72.

00:08:53.289 --> 00:08:56.464
So, let's add 72 to both
sides of this equation.

00:08:56.464 --> 00:08:59.029
So, plus 72.

00:08:59.029 --> 00:09:03.429
We get on the left hand side, AN plus BM,

00:09:03.429 --> 00:09:06.564
these cancel out, is equal to,

00:09:06.564 --> 00:09:11.564
so six plus two is eight,
seven plus seven is 14,

00:09:12.017 --> 00:09:14.684
you have a one up there,
one plus two is three.

00:09:14.684 --> 00:09:16.989
So, it's 348.

00:09:16.989 --> 00:09:18.963
So, the eighth term in our sequence,

00:09:18.963 --> 00:09:20.857
which we figured out, which we figured out

00:09:20.857 --> 00:09:25.090
was this business over
here, it is equal to AB.

00:09:25.090 --> 00:09:28.821
Which is the same thing as 1440.

00:09:28.821 --> 00:09:32.563
Plus, plus this thing over here.

00:09:32.563 --> 00:09:34.950
Plus this thing, which
is just the same thing

00:09:34.950 --> 00:09:38.651
as seven times, let me write it up here.

00:09:38.658 --> 00:09:40.448
This thing over here in
blue is the same thing

00:09:40.448 --> 00:09:43.927
as seven times AN plus BM.

00:09:43.927 --> 00:09:46.324
Now, we know that AN plus BM is 348,

00:09:46.324 --> 00:09:50.354
so it's seven times 348.

00:09:50.354 --> 00:09:54.694
And then finally, finally, plus, plus

00:09:54.694 --> 00:09:58.157
this thing over here, plus 49 times MN.

00:09:58.157 --> 00:10:01.533
49, or we could say minus 49 times 72.

00:10:01.533 --> 00:10:03.022
So, let me just, so what this is,

00:10:03.022 --> 00:10:07.662
you could say 49 times negative 72.

00:10:07.662 --> 00:10:10.587
This is going to be the
eighth term in our sequence,

00:10:10.587 --> 00:10:14.689
so let's figure out, let's
figure out what this thing,

00:10:14.689 --> 00:10:16.764
what this thing even is.

00:10:16.764 --> 00:10:19.136
So, let's see how we can,
let's see how we can do it.

00:10:19.136 --> 00:10:20.632
Well, we could just multiply it out.

00:10:20.632 --> 00:10:22.925
So, seven times 348.

00:10:22.925 --> 00:10:24.719
I have no calculator at my disposal,

00:10:24.719 --> 00:10:26.796
seven times 348.

00:10:26.796 --> 00:10:31.225
Eight times seven is 56.

00:10:31.225 --> 00:10:34.627
Four times seven is 28 plus three is,

00:10:34.627 --> 00:10:37.394
or, plus five is 33.

00:10:37.394 --> 00:10:41.852
Three times seven is 21 plus three is 24.

00:10:41.852 --> 00:10:45.094
So this is 2436.

00:10:45.094 --> 00:10:48.493
If we add that to 1440,
so, if you add it to,

00:10:48.493 --> 00:10:49.294
did I do that right?

00:10:49.294 --> 00:10:51.654
Seven times eight is 56,
seven times four is 28,

00:10:51.654 --> 00:10:53.417
plus five is 33.

00:10:53.417 --> 00:10:54.742
Yup, that's right.

00:10:54.742 --> 00:10:57.222
So then we add the 1440 over here.

00:10:57.222 --> 00:11:02.004
We get six seven eight three, 3876.

00:11:02.004 --> 00:11:04.087
So, that's this whole part over here.

00:11:04.087 --> 00:11:05.828
That's all of this business.

00:11:05.828 --> 00:11:07.892
And then we have to take 49 times 72

00:11:07.892 --> 00:11:09.625
and then subtract it from this.

00:11:09.625 --> 00:11:14.192
So, let's take 49 times 72.

00:11:14.192 --> 00:11:17.564
Two times nine is 18,
two times four is eight,

00:11:17.564 --> 00:11:20.154
plus one is nine, stick a zero here.

00:11:20.154 --> 00:11:24.064
Nine times seven is 63.

00:11:24.064 --> 00:11:29.064
Seven times four is 28 plus six is 34.

00:11:29.952 --> 00:11:34.952
34, so this is eight, 12,
carry the one, five, and three.

00:11:35.471 --> 00:11:39.777
So, we're gonna subtract 3528.

00:11:39.777 --> 00:11:42.759
So, we get, what do we get here?

00:11:42.759 --> 00:11:46.028
So, this becomes a 16, this becomes a six,

00:11:46.028 --> 00:11:47.986
16 minus eight is eight.

00:11:47.986 --> 00:11:50.761
Six minus two is four.

00:11:50.761 --> 00:11:52.562
Eight minus five is three and then

00:11:52.562 --> 00:11:54.288
three minus three is zero.

00:11:54.288 --> 00:11:56.689
So, the eighth term in this sequence,

00:11:56.689 --> 00:11:58.562
the eighth term in this sequence

00:11:58.562 --> 00:12:02.647
is going to be, I just
wrote it down, it is 348.

00:12:03.467 --> 00:12:05.685
348.

00:12:05.685 --> 00:12:06.987
And we're done.


2003 AIME II problem 7

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.460
Find the area of
rhombus ABCD given

00:00:04.460 --> 00:00:09.150
that the radii of the circles
circumscribed about triangles

00:00:09.150 --> 00:00:15.070
ABD and ACD are 12.5
and 25, respectively.

00:00:15.070 --> 00:00:18.380
So let's draw
ourselves rhombus ABCD.

00:00:18.380 --> 00:00:21.260
So let's draw a rhombus.

00:00:21.260 --> 00:00:22.760
So let me draw it.

00:00:22.760 --> 00:00:24.600
Here we go.

00:00:24.600 --> 00:00:26.620
That's a decent rhombus
right over there.

00:00:26.620 --> 00:00:30.520
We know that all the sides
of a rhombus are equal.

00:00:30.520 --> 00:00:32.800
And let's label the vertices.

00:00:32.800 --> 00:00:39.370
So vertex A, B, C, D. So
there we go, rhombus ABCD.

00:00:39.370 --> 00:00:41.110
And then they say the
radii of the circle

00:00:41.110 --> 00:00:43.740
circumscribed about
triangles ABD--

00:00:43.740 --> 00:00:49.530
so triangle ABD, that's A,
B, D. That is triangle ABD.

00:00:49.530 --> 00:00:51.180
So let's draw out the circle.

00:00:51.180 --> 00:00:52.980
Let's draw-- its circumcircled.

00:00:52.980 --> 00:00:57.020
It's circumscribed
circle, or the circle

00:00:57.020 --> 00:01:01.170
that passes through the
vertices A, B, and D. So

00:01:01.170 --> 00:01:04.519
let me do my best job at that.

00:01:04.519 --> 00:01:08.860
So it would look something--
this is not a trivial thing

00:01:08.860 --> 00:01:09.360
to do.

00:01:09.360 --> 00:01:11.460
It's not always easy.

00:01:11.460 --> 00:01:14.710
So let's do it like that.

00:01:14.710 --> 00:01:15.410
There we go.

00:01:15.410 --> 00:01:18.280
That's its circumscribed
circle, or it's

00:01:18.280 --> 00:01:23.000
the circle circumscribed
about ABD right over there.

00:01:23.000 --> 00:01:28.260
Now they're telling us
that its diameter is 12.5--

00:01:28.260 --> 00:01:30.510
so they're saying that this
diameter right over here,

00:01:30.510 --> 00:01:33.150
so if I were to draw a diameter
of this circle right over here,

00:01:33.150 --> 00:01:36.250
it is 12.5.

00:01:36.250 --> 00:01:40.730
Now, the other circle-- the
circumcircle for triangle ACD,

00:01:40.730 --> 00:01:41.480
so let's draw ACD.

00:01:47.070 --> 00:01:50.320
So let's draw a circle that can
go through these three points.

00:01:50.320 --> 00:01:54.585
It looks like it would have
to be something like this.

00:01:57.640 --> 00:02:00.750
It looks like it would have to
be a somewhat bigger circle,

00:02:00.750 --> 00:02:03.250
and that gels with
the information

00:02:03.250 --> 00:02:07.150
that they gave us,
the way I drew it.

00:02:07.150 --> 00:02:09.889
So the circle would look
something like that.

00:02:09.889 --> 00:02:11.835
I don't want to
spend too much time

00:02:11.835 --> 00:02:12.960
trying to draw that circle.

00:02:12.960 --> 00:02:16.651
But they're telling us that
it's-- I should be very

00:02:16.651 --> 00:02:17.150
careful.

00:02:17.150 --> 00:02:19.960
They're saying that the radius
is 12.5, not the diameter.

00:02:19.960 --> 00:02:21.362
So let me make it very clear.

00:02:21.362 --> 00:02:22.820
Actually, let me
delete that circle

00:02:22.820 --> 00:02:24.682
since it's just so messy.

00:02:24.682 --> 00:02:28.040
And I can delete that 12.5 too.

00:02:28.040 --> 00:02:30.750
Let me get-- there you go.

00:02:30.750 --> 00:02:34.900
So the 12.5 is the radius,
the radii of the circle.

00:02:34.900 --> 00:02:38.710
So this first circle around
ABD-- around triangle ABD--

00:02:38.710 --> 00:02:41.030
this distance right
over here is 12.5.

00:02:41.030 --> 00:02:45.310
This distance over
here is also 12.5.

00:02:45.310 --> 00:02:47.475
Now, let's focus
on triangle ACD.

00:02:53.770 --> 00:02:55.500
Let's focus on that triangle.

00:02:55.500 --> 00:03:03.630
Its circumcircle will
look something like this.

00:03:03.630 --> 00:03:05.500
Let me draw.

00:03:05.500 --> 00:03:07.790
There you-- oh, that
doesn't look too good.

00:03:07.790 --> 00:03:08.840
There you go.

00:03:08.840 --> 00:03:10.186
Something like that.

00:03:10.186 --> 00:03:12.560
The whole point here isn't
trying to draw a circumcircle.

00:03:12.560 --> 00:03:16.040
But it's a circle that will
go through those three points.

00:03:19.240 --> 00:03:21.980
And it has a radius of 25.

00:03:21.980 --> 00:03:27.100
So if you had its center, if I
were to draw a diameter of it,

00:03:27.100 --> 00:03:28.610
it is 25.

00:03:28.610 --> 00:03:29.610
Fair enough.

00:03:29.610 --> 00:03:35.647
Now, we would need to figure out
the area of rhombus ABC and D.

00:03:35.647 --> 00:03:38.230
Now, if you've been seeing the
videos that I've been uploading

00:03:38.230 --> 00:03:39.800
lately, I've actually
been uploading

00:03:39.800 --> 00:03:42.340
a few of the
prerequisites for this.

00:03:42.340 --> 00:03:45.420
Because there is a formula,
and we proved the formula

00:03:45.420 --> 00:03:48.520
in the geometry and the
competition math playlist.

00:03:48.520 --> 00:03:53.910
We proved the formula that
relates the area of a triangle

00:03:53.910 --> 00:03:57.550
to the radius of
its circumcircle.

00:03:57.550 --> 00:04:00.270
And let me just rewrite the
formula right over here.

00:04:00.270 --> 00:04:05.170
The formula is the radius
of a triangle's circumcircle

00:04:05.170 --> 00:04:10.805
is equal to the product
of the triangle's sides.

00:04:15.150 --> 00:04:18.140
All of that over 4 times
the area of the triangle.

00:04:18.140 --> 00:04:20.959
So let's see if we can use this
formula that we have proved

00:04:20.959 --> 00:04:25.490
in a previous video to figure
out areas of triangle ABD--

00:04:25.490 --> 00:04:28.830
or express them somehow--
and triangle ACD and then

00:04:28.830 --> 00:04:30.590
see if we can use
that information

00:04:30.590 --> 00:04:33.950
to figure out the area
of the entire rhombus.

00:04:33.950 --> 00:04:37.530
So let me redraw it a
little bit because I

00:04:37.530 --> 00:04:40.286
think my diagram's
gotten kind of messy.

00:04:40.286 --> 00:04:41.410
So I'll redraw the rhombus.

00:04:44.340 --> 00:04:46.700
We actually won't even have
to draw the circumcircles,

00:04:46.700 --> 00:04:49.180
or the circumscribed
circles, because we

00:04:49.180 --> 00:04:51.710
know this formula
right over here.

00:04:51.710 --> 00:04:57.980
So this is A, B, C,
and D. Now, let's

00:04:57.980 --> 00:04:59.970
think first about triangle ABD.

00:05:05.160 --> 00:05:07.230
Actually, let me just
draw the diagonals here.

00:05:07.230 --> 00:05:08.650
BD is one of the diagonals.

00:05:08.650 --> 00:05:10.280
AC is another one
of the diagonals.

00:05:12.900 --> 00:05:14.790
We know that the
diagonals of a rhombus

00:05:14.790 --> 00:05:16.270
are perpendicular bisectors.

00:05:16.270 --> 00:05:17.760
We know that that's
a right angle.

00:05:17.760 --> 00:05:18.470
That's a right angle.

00:05:18.470 --> 00:05:19.345
That's a right angle.

00:05:19.345 --> 00:05:20.430
That's a right angle.

00:05:20.430 --> 00:05:24.930
And we know that this length
is equal to this length,

00:05:24.930 --> 00:05:30.300
and we also know that that
length is equal to that length.

00:05:30.300 --> 00:05:33.920
Now, if we knew this green
length here or this blue length

00:05:33.920 --> 00:05:37.400
here, we would be able to figure
out the area of the rhombus.

00:05:37.400 --> 00:05:40.680
Let's label them.

00:05:40.680 --> 00:05:44.650
Let's call this lowercase a,
and let's call this length

00:05:44.650 --> 00:05:48.390
over here lowercase
b. a times b times 1/2

00:05:48.390 --> 00:05:51.630
would be the area of this
triangle right over there.

00:05:51.630 --> 00:05:54.010
a times b times 1/2
times 2 would give us

00:05:54.010 --> 00:05:55.530
this area and that area.

00:05:55.530 --> 00:05:56.970
Or another way to
think about it,

00:05:56.970 --> 00:05:59.000
this triangle is
completely congruent.

00:05:59.000 --> 00:06:01.780
It has sides a, b, and
this side over here.

00:06:01.780 --> 00:06:04.810
All of these four triangles
have those three sides,

00:06:04.810 --> 00:06:07.570
so all four of these
triangles are congruent.

00:06:07.570 --> 00:06:10.460
So you could take the
area of this triangle,

00:06:10.460 --> 00:06:13.620
multiply it by 4-- you have
the area of the rhombus.

00:06:13.620 --> 00:06:16.920
Let me write this down.

00:06:16.920 --> 00:06:23.480
The rhombus area is
equal to 4 times 1/2ab.

00:06:23.480 --> 00:06:28.030
1/2ab gives us just this
triangle right over here.

00:06:28.030 --> 00:06:32.780
4 times that, so 4
times 1/2ab is 2ab,

00:06:32.780 --> 00:06:34.550
is going to be the
area of the rhombus.

00:06:34.550 --> 00:06:38.590
If we can somehow
figure out a and b,

00:06:38.590 --> 00:06:40.990
we can figure out
the rhombus' area.

00:06:40.990 --> 00:06:45.330
So let's focus on this
first piece of information.

00:06:45.330 --> 00:06:50.840
Let's focus on
triangle AB and D. Now

00:06:50.840 --> 00:06:54.290
they tell us that its
circumradius is 12.5.

00:06:54.290 --> 00:06:56.400
So let's just use this
formula right over here.

00:06:56.400 --> 00:07:00.330
We get 12.5.

00:07:00.330 --> 00:07:03.260
It's circumradius is 12.5
is equal to the product

00:07:03.260 --> 00:07:05.240
of the length of the sides.

00:07:05.240 --> 00:07:07.660
So what's the length
of the sides here?

00:07:07.660 --> 00:07:10.520
So we have this side
right over here, side bd.

00:07:10.520 --> 00:07:12.860
That's just going
to be 2a, right?

00:07:12.860 --> 00:07:14.700
That's an a plus
another a, so it's

00:07:14.700 --> 00:07:18.930
going to be 2a, times
this side right over here.

00:07:18.930 --> 00:07:21.740
What's this side,
which is just one

00:07:21.740 --> 00:07:24.090
of the sides of the rhombus.

00:07:24.090 --> 00:07:26.800
Well, this is the hypotenuse
of this right triangle

00:07:26.800 --> 00:07:27.970
right over here, right?

00:07:27.970 --> 00:07:29.420
This is a right angle.

00:07:29.420 --> 00:07:31.890
So it's going to be the
square root of a squared

00:07:31.890 --> 00:07:33.650
plus b squared.

00:07:33.650 --> 00:07:35.400
But all of the sides
are going to be that.

00:07:35.400 --> 00:07:36.210
It's a rhombus.

00:07:36.210 --> 00:07:39.650
All the sides are the same--
a squared plus b squared.

00:07:39.650 --> 00:07:43.110
They're all going to have
that exact same length.

00:07:43.110 --> 00:07:45.360
So the product of
the sides-- you

00:07:45.360 --> 00:07:49.710
have 2a-- that's the length of
bd-- times the length of ba,

00:07:49.710 --> 00:07:52.890
which is going to be the
square root of a squared

00:07:52.890 --> 00:07:56.380
plus b squared, times
the length of ad,

00:07:56.380 --> 00:08:01.350
which is the square root of
a squared plus b squared.

00:08:01.350 --> 00:08:08.400
All of that over
4 times the area.

00:08:08.400 --> 00:08:10.565
4 times the area of ABD.

00:08:13.350 --> 00:08:16.520
Now, what's the
area of a, b, and d?

00:08:16.520 --> 00:08:20.150
Well, ABD is just two of these
triangles right over here.

00:08:20.150 --> 00:08:22.780
This guy right
over here is 1/2ab.

00:08:22.780 --> 00:08:26.110
This guy over here
is also 1/2ab.

00:08:26.110 --> 00:08:29.380
So the entire area is going
to be two of these guys,

00:08:29.380 --> 00:08:32.330
so it's just going
to be a times b.

00:08:32.330 --> 00:08:34.690
It gives you the area of
both of these triangles.

00:08:34.690 --> 00:08:36.230
Each of them are 1/2ab.

00:08:36.230 --> 00:08:41.697
So instead of writing area
right here, I could write ab.

00:08:41.697 --> 00:08:42.280
Now let's see.

00:08:42.280 --> 00:08:47.724
This simplifies to
12.5, is equal to.

00:08:47.724 --> 00:08:49.640
Divide the numerator and
the denominator by 2.

00:08:49.640 --> 00:08:50.473
So that becomes a 1.

00:08:50.473 --> 00:08:52.090
That becomes a 2, divided by a.

00:08:52.090 --> 00:08:53.480
That becomes a 1.

00:08:53.480 --> 00:08:54.436
That becomes a 1.

00:08:54.436 --> 00:08:56.310
Square root of a squared
plus b squared times

00:08:56.310 --> 00:08:57.935
square root of a
squared plus b squared

00:08:57.935 --> 00:09:01.710
is just a squared
plus b squared.

00:09:01.710 --> 00:09:03.885
And the denominator,
we're just left with a 2b.

00:09:06.410 --> 00:09:08.270
So this first piece
of information,

00:09:08.270 --> 00:09:13.310
the circumradius
for ABD being 12.5,

00:09:13.310 --> 00:09:15.970
gives us this equation
right over here.

00:09:15.970 --> 00:09:18.500
Now, let's do the same
thing for triangle ACD.

00:09:25.460 --> 00:09:28.670
It's circumradius is 25.

00:09:28.670 --> 00:09:34.520
25 is equal to the
length of this side.

00:09:34.520 --> 00:09:35.370
This is a b.

00:09:35.370 --> 00:09:38.205
This is also a b, so
it's going to be 2b.

00:09:38.205 --> 00:09:40.470
2b times the length
of this side,

00:09:40.470 --> 00:09:44.380
which is just the square root
of a squared plus b squared,

00:09:44.380 --> 00:09:46.900
times the length of this
side, which is, again,

00:09:46.900 --> 00:09:52.850
just the square root of
a squared plus b squared.

00:09:52.850 --> 00:09:56.930
All of that over
4 times the area.

00:09:56.930 --> 00:09:59.750
Now, the area, once
again-- it's this triangle,

00:09:59.750 --> 00:10:02.040
which is 1/2ab, plus
this triangle, which

00:10:02.040 --> 00:10:03.210
is another 1/2ab.

00:10:03.210 --> 00:10:05.462
You add them together,
you just get ab.

00:10:05.462 --> 00:10:10.402
You just get ab, 2 divided
by 2, you get a 1 there.

00:10:10.402 --> 00:10:11.110
You get a 2 here.

00:10:11.110 --> 00:10:12.940
Divide by b, get a 1.

00:10:12.940 --> 00:10:14.600
That just becomes an a.

00:10:14.600 --> 00:10:19.770
And so you get 25 is
equal to the numerator,

00:10:19.770 --> 00:10:21.700
square root of a
squared b squared

00:10:21.700 --> 00:10:24.890
times itself is just going
to be a squared plus b

00:10:24.890 --> 00:10:29.970
squared over 2a.

00:10:29.970 --> 00:10:33.170
So that second triangle,
its circumradius being 25

00:10:33.170 --> 00:10:35.174
gives us this equation
right over here.

00:10:35.174 --> 00:10:36.340
Now we can use both of this.

00:10:36.340 --> 00:10:38.130
We have two equations
with two unknowns.

00:10:38.130 --> 00:10:39.280
Let's solve for a and b.

00:10:39.280 --> 00:10:42.230
If we know a and b, we can then
go back here and figure out

00:10:42.230 --> 00:10:44.930
the rhombus's area.

00:10:44.930 --> 00:10:50.720
So over here, we get-- let's
multiply both sides by 2b.

00:10:50.720 --> 00:10:57.420
We get 25b is equal to a
squared plus b squared.

00:10:57.420 --> 00:11:00.480
Over here, if we multiply
both sides by 2a,

00:11:00.480 --> 00:11:06.420
we get 50a is equal to a
squared plus b squared.

00:11:06.420 --> 00:11:08.850
So 50a is equal to a
squared plus b squared.

00:11:08.850 --> 00:11:11.250
25b is equal to a
squared plus b squared.

00:11:11.250 --> 00:11:13.380
So 25b must be the
same thing as 50a.

00:11:13.380 --> 00:11:15.610
They're both a squared
plus b squared.

00:11:15.610 --> 00:11:22.090
So we get 25b must
be equal to 50a.

00:11:22.090 --> 00:11:24.800
They're both equal to a
squared plus b squared.

00:11:24.800 --> 00:11:34.230
Now, divide both sides by
25, you get b is equal to 2a.

00:11:34.230 --> 00:11:36.377
b is equal to--
actually I wanted

00:11:36.377 --> 00:11:37.460
to do that in the magenta.

00:11:40.840 --> 00:11:42.800
b is equal to 2a.

00:11:42.800 --> 00:11:46.080
So we can take this information
and then now substitute back

00:11:46.080 --> 00:11:50.614
into either one of these
equations to solve for b,

00:11:50.614 --> 00:11:51.780
and then we can solve for a.

00:11:51.780 --> 00:11:53.380
So let's go back into this one.

00:11:53.380 --> 00:11:57.220
So we get 50a-- actually
we'll solve for a first.

00:11:57.220 --> 00:12:01.200
50a is equal to a
squared plus b squared.

00:12:01.200 --> 00:12:04.350
Instead of writing b squared, we
know b is the same thing as 2a,

00:12:04.350 --> 00:12:08.040
so let's write 2a squared.

00:12:08.040 --> 00:12:17.650
So we get 50a is equal to
a squared plus 4a squared.

00:12:17.650 --> 00:12:25.030
Or we get 50a is
equal to 5a squared.

00:12:25.030 --> 00:12:28.650
We could divide
both sides by 5a.

00:12:28.650 --> 00:12:32.210
If we divide this
side by 5a, we get 10.

00:12:32.210 --> 00:12:34.300
And if we divide this
side by 5a, we get a.

00:12:34.300 --> 00:12:36.469
So a is equal to 10.

00:12:36.469 --> 00:12:38.260
And then we could just
substitute back here

00:12:38.260 --> 00:12:39.050
to figure out b.

00:12:39.050 --> 00:12:43.150
2 times a is equal to b.
b is equal to 2 times 10,

00:12:43.150 --> 00:12:45.490
which is equal to 20.

00:12:45.490 --> 00:12:48.054
So we know a is 10. b is 20.

00:12:48.054 --> 00:12:49.470
We just have to
go right back here

00:12:49.470 --> 00:12:51.710
to figure out the
area of the rhombus.

00:12:51.710 --> 00:12:55.060
The area of the rhombus
is equal to 2 times-- a

00:12:55.060 --> 00:13:00.120
was 10-- 2 times 10 times 20.

00:13:00.120 --> 00:13:03.870
This is 20 times 20.

00:13:03.870 --> 00:13:07.340
This is equal to 400.

00:13:07.340 --> 00:13:08.290
And we're done.

00:13:08.290 --> 00:13:12.390
The area of rhombus ABCD is 400.


Rhombus diagonals

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.470
I want to do a quick
argument, or proof,

00:00:02.470 --> 00:00:06.270
as to why the diagonals of
a rhombus are perpendicular.

00:00:06.270 --> 00:00:10.000
So remember, a rhombus
is just a parallelogram

00:00:10.000 --> 00:00:12.200
where all four sides are equal.

00:00:12.200 --> 00:00:14.370
In fact, if all four
sides are equal,

00:00:14.370 --> 00:00:16.550
it has to be a parallelogram.

00:00:16.550 --> 00:00:19.410
And just to make things
clear, some rhombuses

00:00:19.410 --> 00:00:20.930
are squares, but
not all of them.

00:00:20.930 --> 00:00:22.680
Because you could have
a rhombus like this

00:00:22.680 --> 00:00:25.580
that comes in where the
angles aren't 90 degrees.

00:00:25.580 --> 00:00:29.457
But all squares are rhombuses,
because all squares,

00:00:29.457 --> 00:00:30.790
they have 90-degree angles here.

00:00:30.790 --> 00:00:33.320
That's not what makes them a
rhombus, but all of the sides

00:00:33.320 --> 00:00:34.080
are equal.

00:00:34.080 --> 00:00:37.660
So all squares are rhombuses,
but not all rhombuses

00:00:37.660 --> 00:00:38.265
are squares.

00:00:38.265 --> 00:00:40.390
Now with that said, let's
think about the diagonals

00:00:40.390 --> 00:00:41.370
of a rhombus.

00:00:41.370 --> 00:00:43.850
And to think about that
a little bit clearer,

00:00:43.850 --> 00:00:46.577
I'm going to draw the rhombus
really as kind of-- I'm

00:00:46.577 --> 00:00:48.160
gonna rotate a little
bit, so it looks

00:00:48.160 --> 00:00:50.270
a little bit like
a diamond shape.

00:00:50.270 --> 00:00:52.710
So notice I'm not
really changing

00:00:52.710 --> 00:00:54.600
any of the properties
of the rhombus.

00:00:54.600 --> 00:00:56.770
I'm just drawing it,
I'm just changing

00:00:56.770 --> 00:00:59.130
its orientation a little bit.

00:00:59.130 --> 00:01:00.960
I'm just changing
its orientation.

00:01:00.960 --> 00:01:04.800
So a rhombus by
definition, the four sides

00:01:04.800 --> 00:01:06.840
are going to be equal.

00:01:06.840 --> 00:01:08.440
Now, let me draw one
of its diagonals.

00:01:08.440 --> 00:01:11.450
And the way I drew it right
here is kind of a diamond.

00:01:11.450 --> 00:01:14.200
One of its diagonals will be
right along the horizontal,

00:01:14.200 --> 00:01:15.720
right like that.

00:01:15.720 --> 00:01:17.650
Now, this triangle on
the top and the triangle

00:01:17.650 --> 00:01:20.289
on the bottom both share
this side, so that side

00:01:20.289 --> 00:01:22.330
is obviously going to be
the same length for both

00:01:22.330 --> 00:01:23.376
of these triangles.

00:01:23.376 --> 00:01:25.250
And then the other two
sides of the triangles

00:01:25.250 --> 00:01:26.400
are also the same thing.

00:01:26.400 --> 00:01:28.350
They're sides of
the actual rhombus.

00:01:28.350 --> 00:01:30.520
So all three sides
of this top triangle

00:01:30.520 --> 00:01:33.300
and this bottom
triangle are the same.

00:01:33.300 --> 00:01:37.610
So this top triangle and this
bottom triangle are congruent.

00:01:37.610 --> 00:01:39.140
They are congruent triangles.

00:01:39.140 --> 00:01:41.940
If you go back to your
ninth grade geometry,

00:01:41.940 --> 00:01:44.590
you'd use the side-side-side
theorem to prove that.

00:01:44.590 --> 00:01:47.410
Three sides are congruent,
then the triangles themselves

00:01:47.410 --> 00:01:48.089
are congruent.

00:01:48.089 --> 00:01:50.380
But that also means that all
the angles in the triangle

00:01:50.380 --> 00:01:51.150
are congruent.

00:01:51.150 --> 00:01:55.120
So the angle that is opposite
this side, this shared side

00:01:55.120 --> 00:01:57.680
right over here
will be congruent

00:01:57.680 --> 00:02:02.110
to the corresponding angle in
the other triangle, the angle

00:02:02.110 --> 00:02:03.190
opposite this side.

00:02:03.190 --> 00:02:05.620
So it would be the
same thing as that.

00:02:05.620 --> 00:02:09.870
Now, both of these triangles
are also isosceles triangles,

00:02:09.870 --> 00:02:12.170
so their base angles are
going to be the same.

00:02:12.170 --> 00:02:14.900
So that's one base angle, that's
the other base angle, right?

00:02:14.900 --> 00:02:16.810
This is an upside down
isosceles triangle,

00:02:16.810 --> 00:02:18.440
this is a right side up one.

00:02:18.440 --> 00:02:21.080
And so if these
two are the same,

00:02:21.080 --> 00:02:23.059
then these are also
going to be the same.

00:02:23.059 --> 00:02:24.850
They're going to be
the same to each other,

00:02:24.850 --> 00:02:26.467
because this is an
isosceles triangle.

00:02:26.467 --> 00:02:28.800
And they're also going to be
the same as these other two

00:02:28.800 --> 00:02:32.860
characters down here, because
these are congruent triangles.

00:02:32.860 --> 00:02:37.160
Now, if we take an altitude,
and actually, I didn't even

00:02:37.160 --> 00:02:39.130
have to talk about
that, since actually,

00:02:39.130 --> 00:02:41.220
I don't think that'll be
relevant when we actually

00:02:41.220 --> 00:02:42.720
want to prove what
we want to prove.

00:02:42.720 --> 00:02:44.970
If we take an altitude
from each of these

00:02:44.970 --> 00:02:47.880
vertices down to this
side right over here.

00:02:47.880 --> 00:02:53.230
So an altitude by
definition is going

00:02:53.230 --> 00:02:55.840
to be perpendicular down here.

00:02:55.840 --> 00:02:59.610
Now, an isosceles triangle
is perfectly symmetrical.

00:02:59.610 --> 00:03:02.002
If you drop an
altitude from the-- I

00:03:02.002 --> 00:03:04.210
guess you could call it the
top, or the unique angle,

00:03:04.210 --> 00:03:07.550
or the unique vertex in
an isosceles triangle--

00:03:07.550 --> 00:03:11.080
you will split it into two
symmetric right triangles.

00:03:11.080 --> 00:03:13.050
Two right triangles
that are essentially the

00:03:13.050 --> 00:03:14.880
mirror images of each other.

00:03:18.600 --> 00:03:22.080
You will also bisect
the opposite side.

00:03:22.080 --> 00:03:25.310
This altitude is, in fact,
a median of the triangle.

00:03:25.310 --> 00:03:27.260
Now we could do it
on the other side.

00:03:27.260 --> 00:03:29.990
The same exact thing
is going to happen.

00:03:33.420 --> 00:03:35.210
We are bisecting
this side over here.

00:03:35.210 --> 00:03:37.634
This is a right angle.

00:03:37.634 --> 00:03:39.050
And so essentially
the combination

00:03:39.050 --> 00:03:42.360
of these two altitudes
is really just a diagonal

00:03:42.360 --> 00:03:43.590
of this rhombus.

00:03:43.590 --> 00:03:47.770
And it's at a right angle to the
other diagonal of the rhombus.

00:03:47.770 --> 00:03:51.484
And it bisects that other
diagonal of the rhombus.

00:03:51.484 --> 00:03:53.650
And we could make the exact
same argument over here.

00:03:53.650 --> 00:03:59.510
You could think of an
isosceles triangle over here.

00:03:59.510 --> 00:04:01.480
This is an altitude of it.

00:04:01.480 --> 00:04:04.630
It splits it into two
symmetric right triangles.

00:04:04.630 --> 00:04:06.750
It bisects the opposite side.

00:04:06.750 --> 00:04:08.840
It's essentially a
median of that triangle.

00:04:08.840 --> 00:04:10.540
Any isosceles triangle.

00:04:10.540 --> 00:04:13.560
Any isosceles triangle, if
that side's equal to that side,

00:04:13.560 --> 00:04:16.324
if you drop an altitude,
these two triangles

00:04:16.324 --> 00:04:17.740
are going to be
symmetric, and you

00:04:17.740 --> 00:04:21.790
will have bisected
the opposite side.

00:04:21.790 --> 00:04:24.470
So by the same argument, that
side's equal to that side,

00:04:24.470 --> 00:04:28.700
so the two diagonals
of any rhombus

00:04:28.700 --> 00:04:32.360
are perpendicular to each other
and they bisect each other.

00:04:34.950 --> 00:04:37.830
Anyway, hopefully you
found that useful.


Area circumradius formula proof

WEBVTT
Kind: captions
Language: en

00:00:00.109 --> 00:00:01.680
What i want to do in this video

00:00:01.680 --> 00:00:03.349
is to come up with a relationship

00:00:03.349 --> 00:00:05.741
between the area of a triangle

00:00:05.741 --> 00:00:08.495
and the triangle's circumscribed circle

00:00:08.495 --> 00:00:09.918
or circum-circle.

00:00:09.918 --> 00:00:11.722
So before we think about the circum-circle

00:00:11.722 --> 00:00:14.476
let's just think about the area of the triangle.

00:00:14.476 --> 00:00:24.806
So let's say that the triangle looks something like this.

00:00:24.806 --> 00:00:27.025
Actually I don't want to make it look isosceles.

00:00:27.025 --> 00:00:31.591
So let me make it a little bit

00:00:31.591 --> 00:00:36.009
so it doesn't look like any particular type of triangle

00:00:36.009 --> 00:00:40.038
and let's call this traingle "ABC".

00:00:40.038 --> 00:00:41.237
That's the vertices

00:00:41.237 --> 00:00:45.268
and then the length of the side opposite "A" is "a"

00:00:45.268 --> 00:00:48.775
"b" over here, and then "c"

00:00:48.775 --> 00:00:51.249
We know how to calculate the area of this triangle

00:00:51.249 --> 00:00:52.587
if we know its height.

00:00:52.587 --> 00:00:57.005
If we drop an altitude right here

00:00:57.005 --> 00:00:59.571
and if this altitude has length "h"

00:00:59.571 --> 00:01:05.280
we know that the area of [ABC]

00:01:05.280 --> 00:01:07.221
- and we write [ABC] with the brackets around it

00:01:07.221 --> 00:01:09.858
means the area of the traingle [ABC] -

00:01:09.858 --> 00:01:15.310
is equal to 1/2 times the base, which is "b"

00:01:15.310 --> 00:01:18.847
times the height.

00:01:18.847 --> 00:01:19.544
Fair enough.

00:01:19.544 --> 00:01:21.616
We have an expression for the area.

00:01:21.616 --> 00:01:22.719
Let's see if we can somehow

00:01:22.719 --> 00:01:25.501
relate some of these things with the area

00:01:25.501 --> 00:01:30.332
to the radius of the triangle's circumscribed circle.

00:01:30.332 --> 00:01:32.654
So the circumscribed circle is a circle

00:01:32.654 --> 00:01:36.424
that passes through all of the vertices of the triangle

00:01:36.424 --> 00:01:38.910
and every triangle has a circumscribed circle.

00:01:38.910 --> 00:01:40.326
So let me try to draw it.

00:01:40.326 --> 00:01:43.667
This is the hard part, right over here

00:01:43.667 --> 00:01:49.825
so it might look something like this

00:01:49.825 --> 00:01:51.812
That's fair enough. That's close enough to a circle

00:01:51.812 --> 00:01:53.800
I think you get the general idea

00:01:53.800 --> 00:01:56.742
That is the circum-circle for this triangle.

00:01:56.742 --> 00:02:00.608
or this triangle's circumscribed circle.

00:02:00.608 --> 00:02:02.366
Let me label it.

00:02:02.366 --> 00:02:13.186
This is the circum-circle for this triangle.

00:02:13.186 --> 00:02:16.743
Now let's think about the center of that circum-circle

00:02:16.743 --> 00:02:19.607
sometimes refer to as the circumcenter.

00:02:19.607 --> 00:02:21.722
So looks like it would be sitting

00:02:21.722 --> 00:02:23.227
I don't know, just eyeballing it

00:02:23.227 --> 00:02:24.532
right on this little "b" here.

00:02:24.532 --> 00:02:27.355
So that's the circum-circle of the circle

00:02:27.355 --> 00:02:30.343
Let's draw a diameter through that circumcircle

00:02:30.343 --> 00:02:33.129
and draw a diameter from vertex "B"

00:02:33.129 --> 00:02:35.970
through that circumcenter.

00:02:35.970 --> 00:02:41.794
So then we go there, and we just keep going over here

00:02:41.794 --> 00:02:46.850
Let's call this point over here "D".

00:02:46.850 --> 00:02:50.566
Now let's create a triangle with vertices A, B, and D.

00:02:50.566 --> 00:02:53.621
So we can just draw another line over here

00:02:53.621 --> 00:02:56.253
and we have triangle ABD

00:02:56.253 --> 00:02:58.509
Now we proved in the geometry play

00:02:58.509 --> 00:03:00.665
- and it's not actually a crazy prove at all -

00:03:00.665 --> 00:03:05.312
that any triangle that's inscribed in a circle

00:03:05.312 --> 00:03:07.650
where one of the sides of the triangle

00:03:07.650 --> 00:03:11.684
is a diameter of the circle

00:03:11.684 --> 00:03:14.723
then that is going to be a right triangle

00:03:14.723 --> 00:03:16.983
and the angle that is going to be 90 degrees

00:03:16.983 --> 00:03:19.253
is the angle opposite the diameter

00:03:19.253 --> 00:03:21.545
So this is the right angle right here.

00:03:21.545 --> 00:03:24.407
You can derive that, pretty straightforward.

00:03:24.407 --> 00:03:32.255
You have this arc here that is 180 degrees.

00:03:32.255 --> 00:03:33.843
because obviously this is a diameter.

00:03:33.843 --> 00:03:36.793
And it subtends this inscribed angle.

00:03:36.793 --> 00:03:39.896
We've also proved that an inscribed angle

00:03:39.896 --> 00:03:41.287
that is subtended by the arc

00:03:41.287 --> 00:03:43.034
will be half of the arc length

00:03:43.034 --> 00:03:45.485
This is an 180 degree arc

00:03:45.485 --> 00:03:47.202
so this is going to be a 90 degree angle.

00:03:47.202 --> 00:03:50.434
So either way this's going to be 90 degrees over there

00:03:50.434 --> 00:03:52.143
The other thing we see

00:03:52.143 --> 00:03:54.516
is that we have this arc right over here

00:03:54.516 --> 00:03:55.670
that I'm drawing in magenta

00:03:55.670 --> 00:03:58.240
the arc that goes from "A" to "B"

00:03:58.240 --> 00:04:01.692
That arc subtends two different angles in our drawing

00:04:01.692 --> 00:04:05.239
- it subtends this angle right over here, angle ACB

00:04:05.239 --> 00:04:07.668
it subtends that right over there -

00:04:07.699 --> 00:04:11.066
but it also subtends angle ADB

00:04:11.066 --> 00:04:12.773
that's why we construct it this way

00:04:12.773 --> 00:04:14.753
So it also subtends this

00:04:14.753 --> 00:04:18.917
So these two angles are going to be congruent.

00:04:18.917 --> 00:04:21.059
They'll both have half the degree measure

00:04:21.059 --> 00:04:22.241
of this arc over here

00:04:22.241 --> 00:04:24.232
because they're both inscribed angles

00:04:24.232 --> 00:04:28.356
subtended by the same exact arc.

00:04:28.356 --> 00:04:30.118
Something interesting is popping up.

00:04:30.118 --> 00:04:31.888
We have two triangles here

00:04:31.888 --> 00:04:40.434
we have triangle ABD and triangle BEC

00:04:40.434 --> 00:04:41.883
They have two angles that resemble

00:04:41.883 --> 00:04:45.747
They have right angle and this magenta angle

00:04:45.747 --> 00:04:47.924
and their third angle must be the same.

00:04:47.924 --> 00:04:49.209
We'll do it in yellow

00:04:49.209 --> 00:04:53.842
The third angle must be congruent to that angle.

00:04:53.842 --> 00:04:55.942
They have three angles that are the same.

00:04:55.942 --> 00:04:59.018
They must be similar triangles.

00:04:59.018 --> 00:05:01.539
or the ratio between the corresponding sides

00:05:01.539 --> 00:05:03.725
must be the same.

00:05:03.725 --> 00:05:05.630
So we can use that information now

00:05:05.630 --> 00:05:07.635
to relate the length of this side

00:05:07.635 --> 00:05:09.841
which is really the diameter, is two times the radius

00:05:09.841 --> 00:05:12.705
to the height of this smaller triangle.

00:05:12.705 --> 00:05:13.645
We know the relationship

00:05:13.645 --> 00:05:14.670
between the height of the smaller triangle

00:05:14.670 --> 00:05:15.659
and the area

00:05:15.659 --> 00:05:18.141
and we essentially are in the home stretch.

00:05:18.141 --> 00:05:19.355
So let's do that

00:05:19.355 --> 00:05:21.341
So these are two similar triangles

00:05:21.341 --> 00:05:33.448
We know that the ratio of C to this diameter right here

00:05:33.448 --> 00:05:34.778
What's the length of the diameter?

00:05:34.778 --> 00:05:38.544
The length of the diameter is 2 times the radius

00:05:38.544 --> 00:05:40.908
This is the radius.

00:05:40.908 --> 00:05:45.270
We know that the ratio, C to two times the radius

00:05:45.270 --> 00:05:52.030
is going to be the same exact thing as the ratio of "h"

00:05:52.030 --> 00:05:53.838
- and we want to make sure we're using the same side -

00:05:53.838 --> 00:05:55.529
to the hypoteneuse of that triangle

00:05:55.529 --> 00:05:59.553
to the ratio of "H" to "A".

00:05:59.553 --> 00:06:01.742
And the way we figured that out

00:06:01.742 --> 00:06:03.263
we look at corresponding sides.

00:06:03.263 --> 00:06:06.641
"C" and the hypoteneuse are both the sides

00:06:06.641 --> 00:06:08.923
adjacent to this angle right over here

00:06:08.923 --> 00:06:11.647
So you have "H" and "A".

00:06:11.647 --> 00:06:15.405
So "C" is to "2r "as "H" is to "a".

00:06:15.405 --> 00:06:19.668
Or, we could do a lot of things.

00:06:19.668 --> 00:06:23.380
1, we could solve for h over here

00:06:23.380 --> 00:06:28.146
and substitute an expression that has the area

00:06:28.146 --> 00:06:30.512
Actually let's just do that

00:06:30.543 --> 00:06:33.811
So if we use this first expression for the area.

00:06:33.811 --> 00:06:39.827
We could multiply both sides by two.

00:06:39.827 --> 00:06:44.711
And divide both sides by B.

00:06:44.711 --> 00:06:47.202
That cancels with that.

00:06:47.202 --> 00:06:50.915
We get that H is equal to 3 times the area over B.

00:06:50.915 --> 00:06:56.992
We can rewrite this relationship as c/2r is equals to h

00:06:56.992 --> 00:07:05.883
which is 2 times the area of our triangle over B

00:07:05.883 --> 00:07:12.118
and then all of that is going to be over A.

00:07:12.118 --> 00:07:15.074
Or, we could rewrite that second part over here

00:07:15.074 --> 00:07:20.270
as two times the area over

00:07:20.270 --> 00:07:21.977
- we're dividing by "b" and then divided by "a",

00:07:21.977 --> 00:07:25.777
that's the same thing as dividing by ab

00:07:25.777 --> 00:07:27.169
So we can ignore this right here.

00:07:27.169 --> 00:07:31.354
So we have c/2r is equals to 2 times the area over ab

00:07:31.354 --> 00:07:33.966
And now we can cross-multiply

00:07:33.966 --> 00:07:49.362
ab times c is going to be equal to 2r times 2abc.

00:07:49.362 --> 00:07:56.060
So that's going to be 4r times the area of our triangle.

00:07:56.060 --> 00:07:58.787
I just cross multiply this times this

00:07:58.787 --> 00:08:01.235
is going to be equal to that times that.

00:08:01.251 --> 00:08:03.503
We know that cross multiplication is just

00:08:03.503 --> 00:08:07.320
multiplying both sides of the equation by 2r

00:08:07.320 --> 00:08:11.025
and multiplying both sides of the equation by ab.

00:08:11.025 --> 00:08:12.141
So we did that on the left hand side

00:08:12.141 --> 00:08:13.607
we also did that on the right hand side

00:08:13.607 --> 00:08:15.955
2r and ab

00:08:15.955 --> 00:08:19.097
obviously that cancels with that, that cancels with that

00:08:19.097 --> 00:08:22.941
So we get ABC is equal to 2r times 2abc.

00:08:22.941 --> 00:08:26.815
Or 4r times the area of our triangle.

00:08:26.815 --> 00:08:27.991
And now we're in the home stretch.

00:08:27.991 --> 00:08:40.717
We divide both sides of this by 4 times the area

00:08:40.717 --> 00:08:41.752
and we're done.

00:08:41.752 --> 00:08:43.698
This cancels with that, that cancels with that

00:08:43.698 --> 00:08:45.676
and we have our relationship

00:08:45.676 --> 00:08:50.045
The radius, or we can call it the circumradius.

00:08:50.045 --> 00:08:52.805
The radius of this triangle's circumscribed circle

00:08:52.805 --> 00:08:55.495
is equal to the product of the side of the triangle

00:08:55.495 --> 00:09:03.410
divided by 4 times the area of the triangle.

00:09:03.410 --> 99:59:59.999
That's a pretty neat result.