2021 AIME II Problems/Problem 12

Problem

A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution 1

We denote by $A$ , $B$ , $C$ and $D$ four vertices of this quadrilateral, such that $AB = 5$ , $BC = 6$ , $CD = 9$ , $DA = 7$ . We denote by $E$ the point that two diagonals $AC$ and $BD$ meet at. To simplify the notation, we denote $a = AE$ , $b = BE$ , $c = CE$ , $d = DE$ . We denote $\theta = \angle AED$ .

First, we write down an equation of the area of the quadrilateral $ABCD$ . We have ${\rm Area} \ ABCD = {\rm Area} \ \triangle ABE + {\rm Area} \ \triangle BCE + {\rm Area} \ \triangle CDE + {\rm Area} \ \triangle DAE = \frac{1}{2} ab \sin \angle AEB + \frac{1}{2} bc \sin \angle BEC + \frac{1}{2} cd \sin \angle CED + \frac{1}{2} da \sin \angle DEA = \frac{1}{2} \left( ab + bc + cd + da \right) \sin \theta$ . Because ${\rm Area} \ ABCD = 30$ , we have $\left( ab + bc + cd + da \right) \sin \theta = 60$ . We index this equation as Eq (1).

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
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2021 AIME II Problems/Problem 12

Problem

Solution 1

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