2006 AMC 10B Problem 23

Problem 23 of 25HarderAlgebra Geometry

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?

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Solution

Split the quadrilateral into two triangles of areas

R

and

S,

so the shaded area is

T=R+S.

Comparing triangles that share an altitude, base ratios give

\tfrac{R}{3}=\tfrac{T+7}{10}

and

\tfrac{S}{7}=\tfrac{T+3}{14}.

Then

T=R+S=3\cdot\tfrac{T+7}{10}+7\cdot\tfrac{T+3}{14},

10T=3(T+7)

+5(T+3)

=8T+36,

giving

T=18.

Thus, the correct answer is D.

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