AMC12 2023 A

AMC12 2023 A · Q11

AMC12 2023 A · Q11. It mainly tests Exponents & radicals, Trigonometry (basic).

What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$?

斜率为 $2$ 和 $\frac{1}{3}$ 的两条直线所形成的锐角的度量是多少度？

(A) 30 30

(B) 37.5 37.5

(D) 52.5 52.5

(E) 60 60

Answer

Correct choice: (C)

正确答案：（C）

Solution

Remind that $\text{slope}=\dfrac{\Delta y}{\Delta x}=\tan \theta$ where $\theta$ is the angle between the slope and $x$-axis. $k_1=2=\tan \alpha$, $k_2=\dfrac{1}{3}=\tan \beta$. The angle formed by the two lines is $\alpha-\beta$. $\tan(\alpha-\beta)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}=\dfrac{2-1/3}{1+2\cdot 1/3}=1$. Therefore, $\alpha-\beta=\boxed{\textbf{(C)} 45^\circ}$.

回忆斜率 $=\dfrac{\Delta y}{\Delta x}=\tan \theta$，其中 $\theta$ 是斜率与 $x$ 轴的夹角。$k_1=2=\tan \alpha$，$k_2=\dfrac{1}{3}=\tan \beta$。两条直线形成的夹角为 $\alpha-\beta$。$\tan(\alpha-\beta)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}=\dfrac{2-1/3}{1+2\cdot 1/3}=1$。因此，$\alpha-\beta=\boxed{\textbf{(C)} 45^\circ}$。

Topics