2005 AMC 10A Problems/Problem 25
Contents
Problem
In we have
,
, and
. Points
and
are on
and
respectively, with
and
. What is the ratio of the area of triangle
to the area of the quadrilateral
?
Solution 1
We have
But , so
Note: If it is hard to understand why , you can use the fact that the area of a triangle equals
. If angle
, we have that
.
Video Solution
Solution 2 (no trig)
We can let .
Since
,
.
So,
.
This means that
.
Thus,
-Conantwiz2023
Solution 3 (trig)
Using this formula:
Since the area of is equal to the area of
minus the area of
,
.
Therefore, the desired ratio is
Note: was not used in this problem.
Solution 4
Let be on
such that
then we have
Since
we have
Thus
and
Finally, after some calculations,
.
~ Nafer
~ LaTeX changes by tkfun
Solution 5
Let the area of triangle ABC be denoted by [ABC] and the area of quadrilateral ABCD be denoted by [ABCD].
Let the area of be
.
and
share a height, and the ratio of their bases are
, so the area of
is
.
Similarly, and
share a height, and the ratio of their bases is
, so the ratio of
. Therefore,
The ratio
which is answer choice
.
~JH. L
See also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
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