2003 AIME II Problem 7

Area of rhombus from circumradii of triangles

Find the area of rhombus ABCD

giving that the radii of the circles circumscribed

about triangles ABD and ACD are 12 5 and 25 respectively

So let's draw ourselves rhombus ABCD

So let's draw a rhombus!

So let me draw it, and here we go!

That's a decent rhombus right over there;

we know that the all sides of a rhombus are equal

So let's name, let's label the vertices, so vertex A, B, C, D

So there we go rhombus ABCD

And then they say the radii of the circles circumscribed

about triangles ABD, so triangle ABD, that's ABD

That is triangle ABD so let's draw the circle,

let's draw it's circumcircle, its circumscribed circle

or the circle that passes through the vertices A, B and D

So, let me draw, do my best job at that

this is not a trivial thing to do, it's not always easy

So let's, let's do it like that there we go, that's it,

It's circumscribed circle or it's the circle,

circle circumscribed about ABD right over there

Now they're telling us, they're telling us that its diameter is 12 5

so they're saying that this diameter right over here

So if I were to draw a diameter of this circle right over here

it is 12 5

Now there other circle, circumcircle for triangle ACD

so let's draw ACD; A, C, A, C, D

So let's draw a circle that can go through the these 3 point

it looks like it would have to be something,

something like this, something like this

It looks like somewhat bigger circle,

and that that gels with the information that they gave us,

the way I drew it

So the circle would look something like that

I don't wanna spend too much trying, time trying to draw that circle

but they're telling us that it's I should be very careful

They're saying that the radius is 12 5 not the diameter,

so there we make it very clear,

Actually let me delete that circle

since it's just so messy

I can delete that 12 5 too, let me get

There you go, so the 12 5 is the radius the radii of the circle,

so this first circle ABD around a triangle ABD

This distance right over is 12 5

and this distance over here is also 12 5

So let's focus on triangle ACD

Let's focus on that triangle

Its circumcircle -- will look something like this,

there you go something like that

The whole point here is we're trying to draw circumcircle

but it's a point it's a circle

that will go through those three points

And it has a radius of 25,

so if you draw if you had its center

or if I were to draw a diameter of it is 25, fair enough

Now we need to figure out the area of rhombus A, B, C and D

Now if you've been seeing the video that I've been uploaded lately,

actually I've been uploading a few of the prerequisites for this

Because there is a formula, we proved the formula in the geometry

and in the competition math playlist

We proved the formula that relates

the area of a triangle to it's to radius of its circumcircle

And let me just rewrite the formula right over here

The formula is the radius of triangles circumcircle

is equal to the product of the triangles,

the product of the triangles' sides all of that over four

times the area of the triangle

So let's see if we can use this formula

then we have proved in a previous video

to figure out the areas of triangle ABD or express them somehow

and triangle ACD

and see if we can use that information

to figure out the area of the entire rhombus

So let me let me redraw it, let me redraw it a lil' bit

cause I think my diagram's got kinda messy

So I'll redraw the rhombus,

if you want you don't have to draw the circumcircles

or the circumscribed circles

because we know this formula right of here

So this is A, B, C and D now let's think first of about triangle ABD,

Actually let me just draw the diagonals here

BD is one of the diagonals AC

is another one of the diagonals AC is the other diagonals

We know that the diagonals of a rhombus are perpendicular bisectors

we know that that's a right angle,

And we know, that this length is equal to this length

and we also know that that length is equal to that length

Now, if we knew this green length here

and this blue length here we would be able

to figure out the area of the rhombus

Let's label them, let's call this let's call this right over

here let's call this lower case a

and let's this length over here lower case b

A times B times one half would be

the area of this triangle right over there

A time B times one half times 2 would give us this area

and to that area or other way to think about it

This triangle is completely congruent;

it has sides AB and this side right over here

All of these four triangles have those three sides

So, all four of these triangles are congruent

so you could take the area of this triangle multiply it by four

you will have the area of the rhombus

Let me write this down, the rhombus the rhombus area

is equal to four times one half AB

One half AB gives us just this triangle

right over here four times that which will be

so four times one half AB is two AB

is gonna be the area of the rhombus

We can tell more we can figure out A and B,

we can figure out A and B we can figure out the rhombus' area

So let's focus on this first piece of information,

let's focus on triangle let's focus on triangle A, B, and D

They tell us that it's circumradius s 12 5

so let's just use this formula right over here we get

12 5 is equal to its circumradius

12 5 is equal to the product of length of the sides

So what's the length of the sides here?

So we have this side right over here side BD

that's just gonna be 2A right?

That's an A so A plus an A

So it's gonna be 2A times this side right over here

What's this side which is one of the sides of the rhombus?

Well that's, this is a hypothesis of this

right triangle right over here, right?

This is a right angle,

so it's going to be the square root of A squared plus B squared

But all of the sides are going to be like that,

it's a rhombus all the sides are the same

A squared plus B squared they're all going

to have the exact same length

So the product of the sides,

you have 2A that's the length of BD times

the length of BA which is going to be the square root of A

squared plus B squared time the length of AD

which is the square root of A squared plus B squared

all of that over four times

the area four times the area of ABD

So all of that is four times,

so what's the area of A, B, and D

Well ABD is just two of these triangles right over here

This guy over here is one half AB

This guy also is one half AB

So the entire the entire area is going to be two

of these guys right over here which is gonna be A times B

A times B it gives you the area

of both of these triangles each of more than one half AB

So, instead of writing the area right here I can write AB

So, let's see this simplifies to 12 5 is equal to

divide the numerator and the denominator by twos

so it becomes one becomes a two becomes a one

That becomes a one, square of A squared plus

B squared times square of A squared plus

B squared is just A squared plus B squared

A squared plus B squared and the denominator were just with 2b

So this first piece of information the circumradius for ABD 12 5

gives us this equation right over here

Now let's just do the same thing for triangle ACD

Its circumradius is 25 is equal to the length of this side

This is a B this is also a B so it's going to be 2B,

2B times the length of the side

which is the square root of A squared plus B squared

times the length of this side which is again

just square root of A square plus B squared

all of that over four times four times the area four times the area

Now the area once again it's just this triangle

which is one half AB plus this triangle which is another one half AB

you add them together you just get AB, you just get AB

Two divided by two you get one there,

you get a two here divide by B, get a one

that just has becomes an A

and so you get 25 is equal to the numerator squared

of A squared of B squared times itself

it's just gonna be A squared plus B squared over 2A

So that second triangle its circumradius being 25

gives us this equation right over here

We can use the both of this;

we have two equations with two unknowns

Let's solve for A and B if we know A and B

we can then go back here to figure out the rhombus' area

So, over here we get over here we get

let's multiply both sided by 2B

we get 25B is equal to A squared plus B squared

Over here if we multiply both sides by 2A

we get 50A is equal to A squared plus B squared

So 50A is equal to A squared plus B squared,

25B is equal to A squared plus B square

so 25B must be the same thing as 50A

they're both A squared plus B squared

So we get 25B must be equal to 50A

they're both equal to A square plus B squared

Now divide both sides no with by 25

you get B is equal to 2A B

I wanna do that in magenta

B is equal to B is equal to 2A

So we can take this information

and then now substitute back into one of these equations

to solve for B so we can solve for A

So let's go back into this one

so we get 50A

I have to solve for A first

50A is equal to A squared plus B squared instead of writing B squared,

we know B is the same thing as 2A

so let's write 2A squared

So we get 50A is equal to A square plus four A squared

or we get 50A is equal to 5A squared divide

both sides we can divide both sides by 5A

If we divided this side by 5 we get 10

and if we divide this side by 5 we get A

So A is equal to 10 and we can just substitute

back here so we can figure out B

Two times A is equal to B,

B is equal to two times 10 which is equal to 20

So we now A is 10 B is 20 we just have to go right back

here to figure out the area of the rhombus

The area of the rhombus is equal to

2 times A was 10, 2 times 10 times 20

This is 20 times 20 this equal to 400 and we're done

The area of the rhombus ABCD is 400