AMC 8 2011 Problem 16

AMC8 2011 年第 16 题

三角形★★★☆☆

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 表示边长为 25，25，30 的三角形的面积，B 表示边长为 25，25，40 的三角形的面积，那么 A 和 B 的关系是什么？

A. $A = \dfrac9{16}B$
B. $A = \dfrac34B$
C.A=B
D. $A = \dfrac43B$
E. $A = \dfrac{16}9B$

正确答案 · Correct Answer

解析 · Solution

Solution 1

The altitude shown divides each triangle into two congruent right triangles. The hypotenuse of each right triangle is 25 25. In △ 𝐴 △A the horizontal leg of each right triangle is 15 15, so the vertical leg is 2 5 2 − 1 5 2 = 20 25 2 −15 2 =20. In △ 𝐵 △B the horizontal leg of each right triangle is 20 20, so the vertical leg is 15 15. The area of △ 𝐴 △A is 1 2 ( 30 ) ( 20 ) = 300 2 1 (30)(20)=300, and the area of △ 𝐵 △B is 1 2 ( 40 ) ( 15 ) = 300 2 1 (40)(15)=300, so the two areas are equal. Answer: 𝐶 C .

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