2021 AIME II Problems/Problem 12
A convex quadrilateral has area and side lengths and in that order. Denote by the measure of the acute angle formed by the diagonals of the quadrilateral. Then can be written in the form , where and are relatively prime positive integers. Find .
We denote by , , and four vertices of this quadrilateral, such that , , , . We denote by the point that two diagonals and meet at. To simplify the notation, we denote , , , . We denote .
First, we use the triangle area formula with sines to write down an equation of the area of the quadrilateral . We have .
Because , we have . We index this equation as Eq (1).
Second, we use the law of cosines to establish four equations for four sides of the quadrilateral .
By applying the law of cosines to , we have . Note that .
Hence, . We index this equation as Eq (2).
Analogously, we can establish the following equation for that (indexed as Eq (3)),
the following equation for that (indexed as Eq (4)),
and the following equation for that (indexed as Eq (5)).
By taking Eq (2) - Eq (3) + Eq (4) - Eq (5) and dividing both sides of the equation by 2, we get . We index this equation as Eq (6).
By taking , we get .
Therefore, by writing this answer in the form of , we have and . Therefore, the answer to this question is .
~ Steven Chen (www.professorchenedu.com)
|2021 AIME II (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|