Perpendicular bisectors
2003 AIME II Problem 7 Area of rhombus from circumradii of triangles
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- 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
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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