Question

设 $A$ 为边长 $25, 25$ 和 $30$ 的三角形面积。设 $B$ 为边长 $25, 25$ 和 $40$ 的三角形面积。 $A$ 与 $B$ 有什么关系？

Let $A$ be the area of the triangle with sides of length $25, 25,$ and $30.$ Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40.$ What is the relationship between $A$ and $B?$

$A = \dfrac{9}{16}B$

$A = \dfrac{3}{4}B$

$A = B$

$A = \dfrac{4}{3}B$

$A = \dfrac{16}{9}B$

答案：C

知识点：等腰三角形三角形面积勾股定理

难度评级：1340

解答：

因为这些三角形是等腰三角形，可以作高，把它们分成如图所示的两个全等直角三角形。

用勾股定理，面积为 $A$ 的三角形的高为类似地，面积为 $B$ 的三角形的高为 $\sqrt{25^2 - 15^2} = 20.$ $\sqrt{25^2 - 20^2} = 15.$

因此同样 $A = \dfrac{1}{2} \cdot 20 \cdot 30 = 300.$ $B = \dfrac{1}{2} \cdot 15 \cdot 40 = 300.$

所以 $A = B$ 。

所以正确答案是 C。

Since these triangles are isosceles, we can drop altitudes to create two congruent right triangles as shown in the diagram.

Using the Pythagorean theorem, we get that the altitude of the triangle with area $A$ equals $\sqrt{25^2 - 15^2} = 20.$ Similarly, we get that the altitude of the triangle with area $B$ equals $\sqrt{25^2 - 20^2} = 15.$

With these altitudes, we can calculate the areas of the triangles. We get that $A = \dfrac{1}{2} \cdot 20 \cdot 30 = 300.$ Similarly, $B = \dfrac{1}{2} \cdot 15 \cdot 40 = 300.$

Therefore, $A = B.$

Thus, C is the correct answer.

2011 AMC 8 第 16 题

答案：C

难度评级：1340

解答：

其他年份的第 16 题