Line 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

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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

The point

(20, 20)

lies on

\ell,

so it also lies on

k;

find the new slope

Rotating a slope

m

45^\circ

gives

\dfrac{m + 1}{1 - m}

Note

(20, 20)

satisfies

3x - 5y + 40 = 0,

so it is on

\ell

and remains on

k.

The slope of

\ell

\tfrac{3}{5}.

Rotating by

45^\circ

gives slope

\dfrac{\tfrac35 + 1}{1 - \tfrac35} = \dfrac{\tfrac85}{\tfrac25} = 4.

Line

k

y - 20 = 4(x - 20).

Setting

y = 0

gives

-20 = 4(x - 20),

x - 20 = -5

and

x = 15.

Thus, B is the correct answer.