Question

Trapezoid $ABCD$ has bases $AB$ and $CD$ and diagonals intersecting at $K.$ Suppose that $AB = 9,$ $DC = 12,$ and the area of $\triangle AKD$ is $24.$ What is the area of trapezoid $ABCD?$

$92$

$94$

$96$

$98$

$100$

Answer: D

Concepts:trapezoid similarity area ratio

Difficulty rating: 1710

Small Hint:

Triangles $AKB$ and $CKD$ are similar with ratio $9 : 12$

Big Hint:

$\triangle AKD$ and $\triangle BKC$ have equal areas; scale the others from $[AKD] = 24$

Solution:

Triangles $AKB$ and $CKD$ are similar with ratio $\dfrac{9}{12} = \dfrac{3}{4}.$

Since $\triangle AKD$ and $\triangle KCD$ have bases $AK$ and $KC$ on the same line and share the same altitude from $D,$ $\dfrac{[KCD]}{[AKD]} = \dfrac{KC}{AK} = \dfrac{4}{3},$ so $[KCD] = 32.$ Similarly $[AKB] = 18.$

Also $[BKC] = [AKD] = 24.$ The total is $24 + 32 + 18 + 24 = 98.$

Thus, the correct answer is D.

2008 AMC 10A Problem 20

Answer: D

Difficulty rating: 1710

Small Hint:

Big Hint:

Solution:

Problem 20 in Other Years