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# 2003 AIME II problem 7

## Video transcript

find the area of rhombus ABCD given 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 here we go that's a decent rhombus right over there we know that all the sides of a rhombus are equal and 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 circle circumscribed about triangles abd so triangle abd that's a B D that is triangle abd so let's draw the circle let's draw this its circumcircle its circumscribed circle or the or the circle that passes through the vertices a B and D so let me draw do my best job at that so it would look something something 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 its circumscribed circle where the 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 the other circle the SIRT the circumcircle for triangle ACD so let's draw a CDA C ay C D so let's draw a circle that can go through these three points it looks like it would have to be something something like this something like this it looks like it would have to be a 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 want to spend too much trying time trying to draw that circle but they're telling us that it's it's I should be very careful they're saying that the radius is 12.5 not the diameter so they'll let me make it very clear and actually delete that circle since it's just so messy I can delete that 12.5 to let me get there you go so the 12.5 is the radius the radii of the circle so this first circle around abd around a triangle abd this distance right over here is 12.5 this distance over here is also is also 12.5 now let's focus on triangle ACD ACD a/c D let's focus on that triangle it's circumcircle will look something like it will look something like this let me draw there you know that doesn't look too good there you go something like that the whole point here is in trying to draw a circumcircle but it's a point it's a circle that will go through those three points through those three points and it has a radius of 25 so if it you draw if you had its Center or if I were to draw a diameter of it it is 25 fair enough now we would need to figure out the area of rhombus a B C and D now if you've been seeing the videos that I've been uploading lately I've actually been uploading a few of the prerequisites for this because there is a formula and we prove the formula and the geometry and the competition math playlist we prove the formula that relates the area of a triangle to its to the radius of its circumcircle and let me just rewrite the formula right over here the formula is the radius of a triangles circumcircle is equal to the product of the triangles the product of the triangles sides all of that over four times the area all of that over four times the area of the triangle so let's see if we can use this formula that we have proved in a previous video to figure out areas of triangle abd or express them somehow and angle a CD and then 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 a little bit because I think my diagrams gotten kind of messy so I'll redraw the rhombus redraw the rhombus we actually won't even have to draw the circum circles or the circumscribed circles because we know this formula right over here so this is a B C and D now let's think first about triangle abd triangle a 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 that's a right angle that's a right angle that's a right angle and we know we know that this length is equal to this length and we also know we also know that that length is equal to that length now if we knew this green length here or this yep 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 lowercase a and let's call this length over here lowercase B a times B times 1/2 would be the area of this triangle right over there a times B times 1/2 times 2 would give us this area and that area or another way to think about it this triangle is completely congruent it has sides a B and this side over here all of these four triangles have those three sides so all four of these triangles are congruent so you can take the area of of this triangle multiply it by four you have the area of the rhombus let me write this down the rhombus the rhombus area is equal to 4 times 1/2 a be 1/2 a B gives us just this triangle right over here 4 times that which will be so 4 times 1/2 a/b is 2 a B is going to be the area of the rhombus so if we can somehow figure out a and B if 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 triangle a B and D now they tell us that it's circumradius is 12.5 so just let's use this formula right over here we get 12.5 12.5 is equal to its circumradius of 12.5 is equal to the product of the lengths of the sides so what's the lengths of the sides here so we have this side right over here side BD that's just going to be 2a right it's that's an A plus another a so it's going to be 2a times this side right over here what's this side what's this side which is just one of the sides of the rhombus well that's this is a the hypotenuse 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 that it's a rhombus all the sides are the same a squared plus B squared they're all going to have that 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 times the length of a D which is the square root of a squared plus B squared all of that all of that over four times four times the area four times the area of a BD so let's call of that over four times now what's the area of a B and D well a BD is just two of these triangles right over here this guy right over here is 1/2 a B this guy over here is also 1/2 a B so the entire the entire area is going to be two of these guys so it's just going to be a times B so a times B it gives you the area of both of these triangles each of them are one-half a B so instead of writing area right here I could write I could write a B now let's see this simplifies to this simplifies to 12 point 5 is equal to divide the numerator and the denominator by 2 so that becomes a 1 that becomes a 2 divided by a that becomes a 1 that becomes a one square root of a squared plus B squared times square root of a squared plus B squared is just a squared plus B squared a squared plus B squared and the denominator we're just left with a - B - B so this first piece of information the circumradius for a BD being 12.5 gives us this equation gives us this equation right over here now let's do the same thing for triangle ACD triangle a a c a CD it's circumradius is 25 25 is equal to is equal to the length of this side this is a B this is also a B so it's going to be 2 B - B times the length of this side which is just the square root of a squared plus B squared times the length of this side which is again just the square root square root of a squared plus B squared all of that over 4 times the area now the area once again it's this triangle which is 1/2 a B plus this triangle which is another 1/2 a B add them together you just get a B you just get a B 2 and divide by 2 you get a 1 there you get a 2 here divided by B get a 1 that just becomes an A and so you get 25 is equal to the numerator square root of a squared B squared times itself is just going to be a squared plus B squared over over 2 over 2a so that second triangle it's circumradius being 25 gives us this equation right over here now we can use 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 and figure out the rhombuses area so over here we get over here we get let's multiply both sides by 2 B we get 25 B is equal to a squared plus B squared over here if we multiply both sides by 2a we get 50 a is equal to a squared plus B squared so fifty a is equal to a squared plus B squared 25 B is equal to a squared plus B squared so 25 B must be the same thing as 50 a they're both a squared plus B squared so we get 25 B must be equal to must be equal to 50 a they're both equal to a squared plus B squared now divide both sides by 25 you get you get B is equal to B is equal to 2a B is equal to which I wanted to do that in the magenta B is equal to B is equal to 2a so we can take this information and then now substitute back into either one of these equations to solve for to solve for B and then we can solve for a so let's go back into this one so we get we get 50 a xu will solve for a first 50 a 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 2a squared so we get 50 50 a is equal to a squared plus a squared plus 4a squared or we get 50 a is equal to 5 a squared divide both sides we could divide both sides by 5 a if we divide this side by 5 a we get 10 and if we divide this side by 5 a we get a so a is equal to 10 and then we could just substitute back here to figure out B 2 times a is equal to B B is equal to 2 times 10 which is equal to 20 so we know 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 is equal to this is equal to 400 and we're done the area of rhombus ABCD is 400