AMC10 2016 B

AMC10 2016 B · Q7

AMC10 2016 B · Q7. It mainly tests Systems of equations, Angle chasing.

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?

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

(A) 75 75

(B) 90 90

(D) 150 150

(E) 270 270

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let $\alpha$ and $\beta$ be the measures of the angles, with $\alpha<\beta$. Then $\dfrac{\beta}{\alpha}=\dfrac{5}{4}$. Because $\alpha<\beta$, it follows that $90^\circ-\beta<90^\circ-\alpha$, so $90^\circ-\alpha=2(90^\circ-\beta)$. This leads to the system of linear equations $4\beta-5\alpha=0$ and $2\beta-\alpha=90^\circ$. Solving the system gives $\alpha=60^\circ$, $\beta=75^\circ$. The requested sum is $\alpha+\beta=135^\circ$.

答案（C）：设$\alpha$和$\beta$为两角的度数，且$\alpha<\beta$。则$\dfrac{\beta}{\alpha}=\dfrac{5}{4}$。由于$\alpha<\beta$，可得$90^\circ-\beta<90^\circ-\alpha$，因此$90^\circ-\alpha=2(90^\circ-\beta)$。这导出线性方程组：$4\beta-5\alpha=0$与$2\beta-\alpha=90^\circ$。解该方程组得$\alpha=60^\circ$，$\beta=75^\circ$。所求和为$\alpha+\beta=135^\circ$。

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