2007 AMC 12A Problems/Problem 13
Problem
A piece of cheese is located at in a coordinate plane. A mouse is at and is running up the line . At the point the mouse starts getting farther from the cheese rather than closer to it. What is ?
Solution 1
The point is the foot of the perpendicular from to the line . The perpendicular has slope , so its equation is . The -coordinate at the foot of the perpendicular satisfies the equation , so and . Thus , and .
Solution 2
If the mouse is at , then the square of the distance from the mouse to the cheese is The value of this expression is smallest when , so the mouse is closest to the cheese at the point , and .
Solution 3
We are trying to find the point where distance between the mouse and is minimized. This point is where the line that passes through and is perpendicular to intersects . By basic knowledge of perpendicular lines, this line is . This line intersects at . So . - MegaLucario1001
Solution 4
If the mouse is at , then the square of the distance from the mouse to the cheese is . The value of this expression is smallest when , so the mouse is closest to the cheese at the point , and . -Paixiao
See also
2007 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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