2020 AMC 12A Problems/Problem 12

Problem

Line $l$ in the coordinate plane has equation $3x-5y+40=0$ . This line is rotated $45\degree$ (Error compiling LaTeX. Unknown error_msg) counterclockwise about the point $(20,20)$ to obtain line $k$ . What is the $x$ -coordinate of the $x$ -intercept of line $k?$

$\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30$

Solution

The slope of the line is $frac{3}{5}$ . We must transform it by $45\degree$ (Error compiling LaTeX. Unknown error_msg). $45\degree$ (Error compiling LaTeX. Unknown error_msg) creates an isosceles right triangle since the sum of the angles of the triangle must be $180\degree$ (Error compiling LaTeX. Unknown error_msg) and one angle is $90\degree$ (Error compiling LaTeX. Unknown error_msg) which means the last leg angle must also be $45\degree$ (Error compiling LaTeX. Unknown error_msg). In the isosceles right triangle, the two legs are congruent. We can, therefore, construct an isosceles right triangle with a line of $frac{3}{5}$ slope on graph paper. That line with $frac{3}{5}$ slope starts at $(0,0)$ and will go to $(5,3)$ , the vector $\opair{5,3}$ (Error compiling LaTeX. Unknown error_msg). Construct another line from $(0,0)$ to $(3,-5)$ , the vector $\opair{3,-5}$ (Error compiling LaTeX. Unknown error_msg). This is $\perp$ and equal to the original line segment. The difference between the two vectors is $\opair{2,8}$ (Error compiling LaTeX. Unknown error_msg), which is the slope $4$ , and that is the slope of line $k$ . Furthermore, the equation $3x-5y+40=0$ passes straight through $(20,20)$ since $3(20)-5(20)+40=60-100+40=0$ , which means that any rotations about $(20,20)$ would contain $(20,20)$ . We can create a line of slope $4$ through $(20,20)$ . The $x$ -intercept is therefore $20-frac{20}{4} = \boxed{\textbf{(B) } 15}.$

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2020 AMC 12A Problems/Problem 12

Problem

Solution