Difference between revisions of "2006 AMC 10B Problems/Problem 23"
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Because the two ratios are equal, we get the equation <math>\frac{A}{6} = \frac {A+10}{10} \Rightarrow 10A = 6A+60 \Rightarrow A = 15</math>. We add the area of triangle <math>EDF</math> to get that the total area of the quadrilateral is <math>\boxed{\textbf{(D) }18}</math>. | Because the two ratios are equal, we get the equation <math>\frac{A}{6} = \frac {A+10}{10} \Rightarrow 10A = 6A+60 \Rightarrow A = 15</math>. We add the area of triangle <math>EDF</math> to get that the total area of the quadrilateral is <math>\boxed{\textbf{(D) }18}</math>. | ||
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== See also == | == See also == |
Revision as of 07:25, 30 August 2019
Contents
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
.
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
.
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.