Difference between revisions of "2008 UNCO Math Contest II Problems/Problem 6"
(Created page with "== Problem == Points <math>A</math> and <math>B</math> are on the same side of line <math>L</math> in the plane. <math>A</math> is <math>5</math> units away from <math>L, B</m...") |
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<asy> | <asy> | ||
− | draw(( | + | draw((-1,0)--(16,0),arrow=Arrow()); |
− | draw(( | + | draw((16,0)--(-1,0),arrow=Arrow()); |
draw((2,0)--(2,5)--(2+sqrt(128),9)--(2+sqrt(128),0),black); | draw((2,0)--(2,5)--(2+sqrt(128),9)--(2+sqrt(128),0),black); | ||
draw((2,5)--(8,0)--(2+sqrt(128),9),dashed); | draw((2,5)--(8,0)--(2+sqrt(128),9),dashed); | ||
Line 20: | Line 20: | ||
MP("5",(2,2.5),W);MP("12",(2+sqrt(128)/2,7),N);MP("9",(2+sqrt(128),4.5),E); | MP("5",(2,2.5),W);MP("12",(2+sqrt(128)/2,7),N);MP("9",(2+sqrt(128),4.5),E); | ||
</asy> | </asy> | ||
− | |||
== Solution == | == Solution == | ||
− | + | Construct <math>B'</math>, the reflection of <math>B</math> across <math>L</math>, and <math>Q</math>, the foot of the perpendicular from <math>A</math> to the segment connecting <math>B</math> to <math>L</math>. By Pythagorean Theorem, <math>AQ^2=AB^2-QB^2=12^2-4^2=128</math>. The minimum possible value of <math>AP+BP</math> is equal to the minimum value of <math>AP+B'P</math>, which is equal to the length of <math>AB'</math>. By Pythagorean Theorem, <math>AB'=\sqrt{AQ^2+B'Q^2}=\sqrt{128+14^2}=\boxed{18}</math>. | |
== See Also == | == See Also == |
Latest revision as of 22:28, 18 March 2018
Problem
Points and
are on the same side of
line
in the plane.
is
units away
from
is
units away from
.
The distance between
and
is
. For
all points
on
what is the smallest
value of the sum
of the distances
from
to
and from
to
?
Solution
Construct , the reflection of
across
, and
, the foot of the perpendicular from
to the segment connecting
to
. By Pythagorean Theorem,
. The minimum possible value of
is equal to the minimum value of
, which is equal to the length of
. By Pythagorean Theorem,
.
See Also
2008 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
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 |