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 use the triangle area formula with sines to 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).

Second, we use the law of cosines to establish four equations for four sides of the quadrilateral $ABCD$ .

By applying the law of cosines to $\triangle AEB$ , we have $a^2 + b^2 - 2 a b \cos \angle AEB = AB^2 = 5^2$ . Note that $\cos \angle AEB = \cos \left( 180^\circ - \theta \right) = \cos \theta$ .

Hence, $a^2 + b^2 + 2 a b \cos \theta = 5^2$ . We index this equation as Eq (2).

Analogously, we can establish the following equation for $\triangle BEC$ that $b^2 + c^2 - 2 b c \cos \theta = 6^2$ (indexed as Eq (3)),

the following equation for $\triangle CED$ that $c^2 + d^2 + 2 c d \cos \theta = 9^2$ (indexed as Eq (4)),

and the following equation for $\triangle DEA$ that $d^2 + a^2 - 2 d a \cos \theta = 7^2$ (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 $\left( ab + bc + cd + da \right) \cos \theta = \frac{21}{2}$ . We index this equation as Eq (6).

By taking $\frac{{\rm Eq} (1)}{{\rm Eq} (6)}$ , we get $\tan \theta = \frac{60}{21/2} = \frac{40}{7}$ .

Therefore, by writing this answer in the form of $\frac{m}{n}$ , we have $m = 40$ and $n = 7$ . Therefore, the answer to this question is $m + n = 40 + 7 = 47$ .

~ Steven Chen (www.professorchenedu.com)

2021 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2021 AIME II Problems/Problem 12

Problem

Solution 1

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