Question

Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0.$ This line is rotated $45^\circ$ counterclockwise about the point $(20, 20)$ to obtain line $k.$ What is the $x$ -coordinate of the $x$ -intercept of line $k?$

$10$

$15$

$20$

$25$

$30$

Answer: B

Concepts:slope transformation coordinate geometry

Difficulty rating: 1630

Solution:

Note $(20, 20)$ satisfies $3x - 5y + 40 = 0,$ so it is on $\ell$ and remains on $k.$ The slope of $\ell$ is $\tfrac{3}{5}.$

Rotating by $45^\circ$ gives slope $\dfrac{\tfrac35 + 1}{1 - \tfrac35} = \dfrac{\tfrac85}{\tfrac25} = 4.$

Line $k$ is $y - 20 = 4(x - 20).$ Setting $y = 0$ gives $-20 = 4(x - 20),$ so $x - 20 = -5$ and $x = 15.$

Thus, B is the correct answer.

2020 AMC 12A Problem 12

Answer: B

Difficulty rating: 1630

Solution:

Problem 12 in Other Years