Difference between revisions of "1985 AIME Problems/Problem 6"
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== Solution 2 == | == Solution 2 == | ||
− | This problem can be done using mass points as well. Assign B a weight of 1 and realize that many of the triangles have the same altitude. After continuously using the formulas | + | This problem can be done using mass points as well. Assign B a weight of 1 and realize that many of the triangles have the same altitude. After continuously using the formulas that state (The sum of the two weights) = (The middle weight), and (The weight <math>\times</math> side) = (Other weight <math>\times</math> the other side), the problem can be trivialized giving the answer <math>315</math> |
Solution by carlos8 | Solution by carlos8 |
Revision as of 13:29, 13 November 2016
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. Assign B a weight of 1 and realize that many of the triangles have the same altitude. After continuously using the formulas that state (The sum of the two weights) = (The middle weight), and (The weight side) = (Other weight
the other side), the problem can be trivialized giving the answer
Solution by carlos8
See also
1985 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |