Difference between revisions of "2007 AMC 12A Problems/Problem 13"
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==Solution 1== | ==Solution 1== | ||
Latest revision as of 17:20, 27 November 2023
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
