Question

三个不同直角三角形的最小角度数之和为 $90^\circ$ 。这三个三角形的边长都是本原勾股数三元组。其中两个是 $3$ - $4$ - $5$ 和 $5$ - $12$ - $13$ 。第三个三角形的周长是多少？

The measures of the smallest angles of three different right triangles sum to $90^\circ.$ All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3$ - $4$ - $5$ and $5$ - $12$ - $13.$ What is the perimeter of the third triangle?

$40$

$126$

$154$

$176$

$208$

答案：C

知识点：三角恒等式勾股数

难度评级：2130

解答：

$3$ - $4$ - $5$ 与 $5$ - $12$ - $13$ 三角形的最小角 $\alpha, \beta$ 满足 $\tan\alpha = \tfrac34$ ， $\tan\beta = \tfrac{5}{12}$ 。 $\begin{aligned} &\tan(\alpha + \beta) = \frac{\tfrac34 + \tfrac{5}{12}}{1 - \tfrac34\cdot\tfrac{5}{12}} \\ &= \frac{\tfrac{14}{12}}{\tfrac{33}{48}} \\ &= \frac{56}{33}. \end{aligned}$

第三个最小角 $\gamma$ 满足 $\gamma = 90^\circ - (\alpha + \beta)$ ，所以 $\tan\gamma = \dfrac{33}{56}$ 。直角边为 $33$ 、 $56$ 的直角三角形斜边为 $\sqrt{33^2 + 56^2} = \sqrt{4225} = 65$ ，构成本原三元组，周长为 $33 + 56 + 65 = 154$ 。

所以正确答案是 C。

The smallest angles $\alpha, \beta$ of the $3$ - $4$ - $5$ and $5$ - $12$ - $13$ triangles have $\tan\alpha = \tfrac34$ and $\tan\beta = \tfrac{5}{12}.$ By the tangent addition formula, $\begin{aligned} &\tan(\alpha + \beta) = \frac{\tfrac34 + \tfrac{5}{12}}{1 - \tfrac34\cdot\tfrac{5}{12}} \\ &= \frac{\tfrac{14}{12}}{\tfrac{33}{48}} \\ &= \frac{56}{33}. \end{aligned}$

The third smallest angle $\gamma$ satisfies $\gamma = 90^\circ - (\alpha + \beta),$ so $\tan\gamma = \dfrac{33}{56}.$ The right triangle with legs $33$ and $56$ has hypotenuse $\sqrt{33^2 + 56^2} = \sqrt{4225} = 65,$ a primitive triple. Its perimeter is $33 + 56 + 65 = 154.$

Thus, the correct answer is C.

2024 AMC 12B 第 21 题

答案：C

难度评级：2130

解答：

其他年份的第 21 题