Difference between revisions of "2006 AMC 10B Problems/Problem 23"
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Since [[triangle]]s <math>AFB</math> and <math>DFB</math> share an [[altitude]] from <math>B</math> and have equal area, their bases must be equal, hence <math>AF=DF</math>. | Since [[triangle]]s <math>AFB</math> and <math>DFB</math> share an [[altitude]] from <math>B</math> and have equal area, their bases must be equal, hence <math>AF=DF</math>. | ||
Revision as of 20:07, 26 April 2019
Contents
[hide]Problem
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7 as shown. What is the area of the shaded quadrilateral?
Solution 1
Label the points in the figure as shown below, and draw the segment . This segment divides the quadrilateral into two triangles, let their areas be
and
.
https://latex.artofproblemsolving.com/a/3/7/a373a9412edc8a46f24525404560b3b355922171.png
Since triangles and
share an altitude from
and have equal area, their bases must be equal, hence
.
Since triangles and
share an altitude from
and their respective bases are equal, their areas must be equal, hence
.
Since triangles and
share an altitude from
and their respective areas are in the ratio
, their bases must be in the same ratio, hence
.
Since triangles and
share an altitude from
and their respective bases are in the ratio
, their areas must be in the same ratio, hence
, which gives us
.
Substituting into the second equation we get
, which solves to
. Then
, and the total area of the quadrilateral is
.
Solution 2
Connect points and
. Triangles
and
share an altitude and their areas are in the ration
. Their bases,
and
, must be in the same
ratio.
Triangles and
share an altitude and their bases are in a
ratio. Therefore, their areas are in a
ratio and the area of triangle
is
.
Triangle and
share an altitude. Therefore, the ratio of their areas is equal to the ratio of bases
and
. The ratio is
where
is the area of triangle
Triangles and
also share an altitude. The ratio of their areas is also equal to the ratio of bases
and
. The ratio is
Because the two ratios are equal, we get the equation . We add the area of triangle
to get that the total area of the quadrilateral is
.
~Zeric Hang
See also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.