In triangle ABC, point D lies on AB and point E lies on AC such that DE is parallel to BC. If AD = 6, DB = 9, and the area of triangle ADE is 24, what is the area of quadrilateral DBCE?

SAT Lines, angles, and triangles Practice Question 2 (Hard)

Question bank / Lines, angles, and triangles

Math · Geometry and TrigonometryHard

In triangle $ABC$ , point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ such that $\overline{DE}$ is parallel to $\overline{BC}$ . If $AD = 6$ , $DB = 9$ , and the area of triangle $ADE$ is $24$ , what is the area of quadrilateral $DBCE$ ?

$36$

$60$

$126$

$150$

Show answer and explanation

Answer: C

Since $\overline{DE} \parallel \overline{BC}$ , triangle $ADE$ is similar to triangle $ABC$ with scale factor $\frac{AB}{AD} = \frac{15}{6} = \frac{5}{2}$ . Areas scale by the square of this factor, so the area of $ABC$ is $24 \cdot \frac{25}{4} = 150$ . The quadrilateral is what remains: $150 - 24 = 126$ .

Choice D is the area of the whole triangle $ABC$ , not the quadrilateral that remains after removing triangle $ADE$ .

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