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

2020 AMC 12A Problem 12 — Solution

2020 AMC 12A Problem 12

Below is the professionally curated solution for Problem 12 of the 2020 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12A solutions, or check the answer key.

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Concepts:slope transformation coordinate geometry

Difficulty rating: 1630

12.

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 $k?$

$10$

$15$

$20$

$25$

$30$

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.

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