2007 AMC 12A Problem 13

Below is the professionally curated solution for Problem 13 of the 2007 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12A solutions, or check the answer key.

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Concepts:coordinate geometry slope

Difficulty rating: 1410

13.

A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18.$ At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a+b?$

$6$

$10$

$14$

$18$

$22$

Solution:

The mouse is closest to the cheese at the foot of the perpendicular from $(12,10)$ to the line.

This perpendicular has slope $\tfrac15,$ so its equation is $y=10+\tfrac15(x-12)=\tfrac15 x+\tfrac{38}{5}.$

Setting $\tfrac15 x+\tfrac{38}{5}=-5x+18$ gives $x=2$ and $y=8.$ Thus $(a,b)=(2,8)$ and $a+b=10.$

Thus, the correct answer is B.

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