AMC10 2007 A

AMC10 2007 A · Q14

AMC10 2007 A · Q14. It mainly tests Triangles (properties), Pythagorean theorem.

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?

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

(A) 8.64 8.64

(B) 12 12

(D) 17.28 17.28

(E) 18 18

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Let the sides of the triangle have lengths \$3x\$, \$4x\$, and \$5x\$. The triangle is a right triangle, so its hypotenuse is a diameter of the circle. Thus \$5x = 2\cdot 3 = 6\$, so \$x = 6/5\$. The area of the triangle is \[ \frac12 \cdot 3x \cdot 4x = \frac12 \cdot \frac{18}{5} \cdot \frac{24}{5} = \frac{216}{25} = 8.64. \] OR A right triangle with side lengths \$3\$, \$4\$, and \$5\$ has area \$(1/2)(3)(4)=6\$. Because the given right triangle is inscribed in a circle with diameter \$6\$, the hypotenuse of this triangle has length \$6\$. Thus the sides of the given triangle are \$6/5\$ as long as those of a \$3-4-5\$ triangle, and its area is \$(6/5)^2\$ times that of a \$3-4-5\$ triangle. The area of the given triangle is \[ \left(\frac{6}{5}\right)^2(6)=\frac{216}{25}=8.64. \]

答案（A）：设三角形的边长分别为 \$3x\$、\$4x\$ 和 \$5x\$。该三角形是直角三角形，因此其斜边是圆的直径。于是 \$5x=2\cdot 3=6\$，所以 \$x=6/5\$。三角形的面积为 \[ \frac12 \cdot 3x \cdot 4x = \frac12 \cdot \frac{18}{5} \cdot \frac{24}{5} = \frac{216}{25} = 8.64. \] 或者边长为 \$3\$、\$4\$、\$5\$ 的直角三角形面积是 \$(1/2)(3)(4)=6\$。由于题给直角三角形内接于直径为 \$6\$ 的圆，所以该三角形的斜边长为 \$6\$。因此题给三角形的各边长度是 \$3-4-5\$ 三角形的 \$6/5\$ 倍，其面积是 \$3-4-5\$ 三角形面积的 \$(6/5)^2\$ 倍。题给三角形的面积为 \[ \left(\frac{6}{5}\right)^2(6)=\frac{216}{25}=8.64. \]

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