Difference between revisions of "2007 AMC 12A Problems/Problem 13"

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==Problem==
 
==Problem==
  
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?
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A piece of cheese is located at <math>(12,10)</math> in a [[coordinate plane]]. A mouse is at <math>(4,-2)</math> and is running up the [[line]] <math>y=-5x+18</math>. At the point <math>(a,b)</math> the mouse starts getting farther from the cheese rather than closer to it. What is  <math>a+b</math>?
 
 
A(6)
 
B(10)
 
C(14)
 
D(18)
 
E(22)
 
  
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<math>\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 14\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 22</math>
  
 
==Solution==
 
==Solution==
  
We are trying to find the point where (12,10) is perpendicular to y=-5x+18. Then the slope of the line that passes through the cheese and (a,b) is 1/5. Therefore, the line is
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We are trying to find the foot of a [[perpendicular]] from <math>(12,10)</math> to <math>y=-5x+18</math>. Then the [[slope]] of the line that passes through the cheese and <math>(a,b)</math> is the negative recipricol of the slope of the line, or <math>\frac 15</math>. Therefore, the line is <math>y=\frac{1}{5}x+\frac{38}{5}</math>. The point where <math>y=-5x+18</math> and <math>y=\frac 15x+\frac{38}5</math> intersect is <math>(2,8)</math>, and <math>2+8=10\ (B)</math>.
* <math>y=\frac{1}{5}x+\frac{38}{5}</math> The point where y=-5x+18 and y=.2x+7.6 intersect is (2,8), and 2+8=10(B).
 
  
 
==See also==
 
==See also==
* [[2007 AMC 12A Problems/Problem 12 | Previous problem]]
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{{AMC12 box|year=2007|ab=A|num-b=12|num-a=14}}
* [[2007 AMC 12A Problems/Problem 14 | Next problem]]
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* [[2007 AMC 12A Problems]]
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[[Category:Introductory Geometry Problems]]

Revision as of 15:28, 10 September 2007

Problem

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 $a+b$?

$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 14\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 22$

Solution

We are trying to find the foot of a perpendicular from $(12,10)$ to $y=-5x+18$. Then the slope of the line that passes through the cheese and $(a,b)$ is the negative recipricol of the slope of the line, or $\frac 15$. Therefore, the line is $y=\frac{1}{5}x+\frac{38}{5}$. The point where $y=-5x+18$ and $y=\frac 15x+\frac{38}5$ intersect is $(2,8)$, and $2+8=10\ (B)$.

See also

2007 AMC 12A (ProblemsAnswer KeyResources)
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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