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2020 AMC 12A Problem 12

Problem 12 of 25IntermediateGeometry

Line ℓ\ell in the coordinate plane has the equation 3x−5y+40=0.3x - 5y + 40 = 0. This line is rotated 45∘45^\circ counterclockwise about the point (20,20)(20, 20) to obtain line k.k. What is the xx-coordinate of the xx-intercept of line k?k?

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Solution

Note (20,20)(20, 20) satisfies 3x−5y+40=0,3x - 5y + 40 = 0, so it is on ℓ\ell and remains on k.k. The slope of ℓ\ell is 35.\tfrac{3}{5}. Rotating by 45∘45^\circ gives slope 35+11−35=8525=4.\dfrac{\tfrac35 + 1}{1 - \tfrac35} = \dfrac{\tfrac85}{\tfrac25} = 4. Line kk is y−20=4(x−20).y - 20 = 4(x - 20). Setting y=0y = 0 gives −20=4(x−20),-20 = 4(x - 20), so x−20=−5x - 20 = -5 and x=15.x = 15. Thus, B is the correct answer.

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Concepts: slope · transformation · coordinate geometry

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