Question

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 $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

答案

Solution:

Label the quadrilateral $ABCD$ with $AB = 5,$ $BC = 6,$ $CD = 9,$ $DA = 7,$ and let the diagonals meet at $P,$ cutting $\overline{AC}$ into $p_1, p_2$ and $\overline{BD}$ into $q_1, q_2.$ With $\varphi = \angle APB,$ the law of cosines in the four corner triangles (whose angles at $P$ alternate between $\varphi$ and $180^\circ - \varphi$ ) gives $BC^2 + DA^2 - AB^2 - CD^2 = 2(p_1q_1 + p_1q_2 + p_2q_1 + p_2q_2)\cos\varphi = 2\,AC \cdot BD \cos\varphi.$

The left side is $36 + 49 - 25 - 81 = -21,$ so $AC \cdot BD\,|\cos\varphi| = \frac{21}{2},$ and the acute angle $\theta$ between the diagonals satisfies $AC \cdot BD \cos\theta = \frac{21}{2}.$ Meanwhile the four corner triangles give the area $\frac{1}{2} AC \cdot BD \sin\theta = 30,$ so $AC \cdot BD \sin\theta = 60.$

Dividing, $\tan\theta = \frac{60}{21/2} = \frac{40}{7},$ so $m + n = 40 + 7 = 47.$

2021 AIME II Problem 12

Solution:

Problem 12 in Other Years