2006 AMC 10B Problem 23

Question

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?

15

17

\dfrac{35}{2}

18

\dfrac{55}{3}

Answer: D

Concepts:area ratio system of equations

Difficulty rating: 1950

Small hint:

Split the quadrilateral into two triangles with areas

R

and

S

Big hint:

Triangles sharing an altitude have areas proportional to their bases

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},

so

10T=3(T+7)

+5(T+3)

=8T+36,

giving

T=18.

Thus, the correct answer is D.