2008 UNCO Math Contest II Problems/Problem 6

Revision as of 15:32, 26 November 2016 by Mathisfun04 (talk | contribs) (Solution)

Problem

Points $A$ and $B$ are on the same side of line $L$ in the plane. $A$ is $5$ units away from $L, B$ is $9$ units away from $L$. The distance between $A$ and $B$ is $12$. For all points $P$ on $L$ what is the smallest value of the sum $AP + PB$ of the distances from $A$ to $P$ and from $P$ to $B$ ?

[asy] draw((-1,0)--(16,0),arrow=Arrow()); draw((16,0)--(-1,0),arrow=Arrow()); draw((2,0)--(2,5)--(2+sqrt(128),9)--(2+sqrt(128),0),black); draw((2,5)--(8,0)--(2+sqrt(128),9),dashed); dot((2,5));dot((8,0));dot((2+sqrt(128),9)); MP("A",(2,5),W);MP("P",(8,0),S);MP("B",(2+sqrt(128),9),E);MP("L",(14,0),S); MP("5",(2,2.5),W);MP("12",(2+sqrt(128)/2,7),N);MP("9",(2+sqrt(128),4.5),E); [/asy]

Solution

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See Also

2008 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
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