Question

两个锐角度数之比为 $5:4$ ，且其中一个角的余角是另一个角余角的两倍。这两个角的度数之和是多少？

The ratio of the measures of two acute angles is $5:4,$ and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?

$75$

$90$

$135$

$150$

$270$

答案：C

知识点：比与比例方程组

难度评级：1200

解答：

设两角为 $\alpha\lt\beta$ ，其中 $\beta=\tfrac54\alpha$ 。较小角的余角较大，所以 $90-\alpha=2(90-\beta)=180-\tfrac52\alpha$ 。由此得到 $\tfrac32\alpha=90$ ，所以 $\alpha=60^\circ$ ， $\beta=75^\circ$ 。两角之和为 $135^\circ$ 。

所以正确答案是 C。

Let the angles be $\alpha\lt\beta$ with $\beta=\tfrac54\alpha.$ The larger complement belongs to the smaller angle, so $90-\alpha=2(90-\beta)=180-\tfrac52\alpha.$ This gives $\tfrac32\alpha=90,$ so $\alpha=60^\circ$ and $\beta=75^\circ.$ The sum is $135^\circ.$

Thus, the correct answer is C.

2016 AMC 12B 第 4 题

答案：C

难度评级：1200

解答：

其他年份的第 4 题