1985 AIME Problems/Problem 6
Contents
Problem
As shown in the figure, triangle is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle
.
Solution
Let the interior point be , let the points on
,
and
be
,
and
, respectively. Let
be the area of
and
be the area of
. Note that
and
share the same altitude from
, so the ratio of their areas is the same as the ratio of their bases. Similarly,
and
share the same altitude from
, so the ratio of their areas is the same as the ratio of their bases. Moreover, the two pairs of bases are actually the same, and thus in the same ratio. As a result, we have:
or equivalently
and so
.
Applying identical reasoning to the triangles with bases and
, we get
so that
and
. Substituting from this equation into the previous one gives
, from which we get
and so the area of
is
.
Solution 2
This problem can be done using mass points as well.
See also
1985 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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