Question

一个边长比为 $3 : 4 : 5$ 的三角形内接于半径为 $3$ 的圆。这个三角形的面积是多少？

A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle of radius $3.$ What is the area of the triangle?

$8.64$

$12$

$5\pi$

$17.28$

$18$

边长比为 $3 : 4 : 5$ 的三角形是直角三角形，所以最长边是圆的直径。

A $3 : 4 : 5$ triangle is right-angled, so its longest side is a diameter

若三边为 $3x, 4x, 5x$ ，则斜边 $5x$ 等于直径 $6$ 。

With sides $3x, 4x, 5x,$ the hypotenuse $5x$ equals the diameter $6$

设三边为 $3x, 4x, 5x$ 。

这个三角形是直角三角形，所以斜边为圆的直径。因此 $5x = 2 \cdot 3 = 6$ ，得到 $x = \tfrac65$ 。

面积为 $\begin{aligned} \tfrac12 (3x)(4x) &= 6x^2 = 6 \cdot \tfrac{36}{25} \\ &= 8.64 \end{aligned}\text{。}$

所以正确答案是 A。

Let the sides be $3x, 4x, 5x.$ The triangle is right-angled, so its hypotenuse is a diameter.

Thus $5x = 2 \cdot 3 = 6,$ giving $x = \tfrac65.$

The area is $\begin{aligned} \tfrac12 (3x)(4x) &= 6x^2 = 6 \cdot \tfrac{36}{25} \\ &= 8.64. \end{aligned}$

Thus, the correct answer is A.