Difference between revisions of "1987 AIME Problems/Problem 9"
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{{AIME box|year=1987|num-b=8|num-a=10}} | {{AIME box|year=1987|num-b=8|num-a=10}} | ||
[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 19:09, 4 July 2013
Problem
Triangle has right angle at
, and contains a point
for which
,
, and
. Find
.
Solution
Let . Since
, each of them is equal to
. By the Law of Cosines applied to triangles
,
and
at their respective angles
, remembering that
, we have
Then by the Pythagorean Theorem, , so
and
See also
1987 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.