2009 AMC 10A Problems/Problem 10
Problem
Triangle has a right angle at
. Point
is the foot of the altitude from
,
, and
. What is the area of
?
Solution
It is a well-known fact that in any right triangle with the right angle at
and
the foot of the altitude from
onto
we have
. (See below for a proof.) Then
, and the area of the triangle
is
.
Proof: Consider the Pythagorean theorem for each of the triangles ,
, and
. We get:
.
Substituting equations 2 and 3 into the left hand side of equation 1, we get .
See Also
2009 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |