2021 AIME II Problems/Problem 12
Problem
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 .
Solution
Since we are asked to find , we can find and separately and then use those values to find . Let us first draw a diagram of this quadrilateral.
[asy] unitsize(4cm); pair A,B,C,D,X; A = (0,0); B = (1,0); C = (1.25,-1); D = (-0.75,-0.75); draw(A--B--C--D--cycle,black+1bp); X = intersectionpoint(A--C,B--D); draw(A--C); draw(B--D); label("",A,NW); abel("",B,NE); label("",C,SE); label("",D,SW); dot(X); label("",X,S); label("",(A+B)/2,N) label("",(B+C)/2,E); label("",(C+D)/2,S); label("",(D+A)/2,W); label("",X,2.5E); label("",(A+X)/2,NE); label("",(B+X)/2,NW); label("",(C+X)/2,SW); label("",(D+X)/2,SE); [/asy]
~ my_aops_lessons
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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