Question

一个凸四边形的面积为 $30$ 边长依次为 $5, 6, 9$ ，和 $7$ ，记该四边形两条对角线所成锐角的度数为 $\theta$ 。则 $\tan \theta$ 可写成 $\frac{m}{n}$ ，其中 $m$ 和 $n$ 是互质正整数。求 $m + n$ 。

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

答案：47

知识点：余弦定理面积三角学

难度评级：2920

解答：

将四边形标记为 $ABCD$ ，其中 $AB = 5$ 、 $BC = 6$ 、 $CD = 9$ 、 $DA = 7$ ，设对角线交于 $P$ ，把 $\overline{AC}$ 分为 $p_1, p_2$ ，把 $\overline{BD}$ 分为 $q_1, q_2$ 。令 $\varphi = \angle APB$ ，在四个角落三角形中使用余弦定理（它们在 $P$ 处的角在 $\varphi$ 与 $180^\circ - \varphi$ 之间交替），得到 $\begin{aligned} &BC^2 + DA^2 - AB^2 - CD^2 \\ &= 2(p_1q_1 + p_1q_2 + p_2q_1 + p_2q_2) \\ &\quad {}\cdot \cos\varphi \\ &= 2\,AC \cdot BD \cos\varphi. \end{aligned}$

左边为 $36 + 49 - 25 - 81 = -21$ ，所以 $AC \cdot BD\,|\cos\varphi| = \frac{21}{2}$ ，而两条对角线所成的锐角 $\theta$ 满足 $AC \cdot BD \cos\theta = \frac{21}{2}$ 。同时，四个角落三角形给出面积 $\frac{1}{2} AC \cdot BD \sin\theta = 30$ ，所以 $AC \cdot BD \sin\theta = 60$ 。

相除得 $\tan\theta = \frac{60}{21/2} = \frac{40}{7}$ ，所以 $m + n = 40 + 7 = 47$ 。

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 $\begin{aligned} &BC^2 + DA^2 - AB^2 - CD^2 \\ &= 2(p_1q_1 + p_1q_2 + p_2q_1 + p_2q_2) \\ &\quad {}\cdot \cos\varphi \\ &= 2\,AC \cdot BD \cos\varphi. \end{aligned}$

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 第 12 题

答案：47

难度评级：2920

解答：

其他年份的第 12 题