2016 AMC 10B 第 7 题

Question

两个锐角的度数之比为

5:4

，且其中一个角的余角是另一个角的余角的两倍。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：比与比例一次方程

难度评级：1070

小提示：

设较小的锐角为

x

。

Let the smaller acute angle be

x

大提示：

较大的角有较小的余角。

The larger angle has the smaller complement

解答：

设较小角为

x

。则较大角为

\frac 54 x

，两角和为

\frac 94 x

。

它们的余角分别是

90-x

与

90 - \frac 54x

。由于

90-x

大于

90-\frac 54 x

，所以

\begin{aligned}90 -x &= 2\left(90 - \frac 54 x\right) \\90-x&= 180-\frac 52 x\\x&=60\end{aligned}\text{。}

因此两角和为

\dfrac 94 \cdot 60=135\text{。}

所以正确答案是 C。

Let the smaller angle be

x.

Then, the larger angle is

\frac 54 x.

Note that their sum is

\frac 94 x.

Their complements are then

90-x

and

90 - \frac 54x.

Since

90-x

is larger than

90-\frac 54 x,

we know

\begin{aligned}90 -x &= 2\left(90 - \frac 54 x\right) \\90-x&= 180-\frac 52 x\\x&=60.\end{aligned}

Therefore, the sum is

\dfrac 94 \cdot 60=135.

Thus, the correct answer is C .

2016 AMC 10B 第 7 题

答案：C

难度评级：1070

小提示：

大提示：

解答：

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