2023 AMC 12A Problem 11

Attempt Problem 11 of the 2023 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

11.

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

2

and

\tfrac13?

30

37.5

45

52.5

60

Use

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

Both the numerator and denominator work out to

\tfrac53

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

45^\circ.

Thus, the correct answer is C.

Problem 11 in other years