2023 AMC 12A 第 11 题

Question

斜率为

2

和

\tfrac13

的两条直线所成锐角的度数是多少？

What is the degree measure of the acute angle formed by lines with slopes

2

and

\tfrac13?

30

37.5

45

52.5

60

使用

\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|

。

Use

\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|

分子和分母都会化为

\tfrac53

。

Both the numerator and denominator work out to

\tfrac53

两直线夹角的正切为

\left|\dfrac{2-\tfrac13}{1+2\cdot\tfrac13}\right| =\left|\dfrac{\tfrac53}{\tfrac53}\right|=1\text{。}

正切为

1

的锐角是

45^\circ

。

所以正确答案是 C。

The tangent of the angle between the lines is

\left|\dfrac{2-\tfrac13}{1+2\cdot\tfrac13}\right| =\left|\dfrac{\tfrac53}{\tfrac53}\right|=1.

The acute angle with tangent

1

is

45^\circ.

Thus, the correct answer is C.

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