Question

In the figure, $\angle EAB$ and $\angle ABC$ are right angles, $AB = 4,$ $BC = 6,$ $AE = 8,$ and $\overline{AC}$ and $\overline{BE}$ intersect at $D.$ What is the difference between the areas of $\triangle ADE$ and $\triangle BDC?$

$2$

$4$

$5$

$8$

$9$

Answer: B

Concepts:triangle area area decomposition

Difficulty rating: 1370

Solution:

Let $x,$ $y,$ and $z$ be the areas of $\triangle ADE,$ $\triangle BDC,$ and $\triangle ABD,$ respectively.

Then $\triangle ABE$ has area $\tfrac12 \cdot 4 \cdot 8 = 16 = x + z,$ and $\triangle ABC$ has area $\tfrac12 \cdot 4 \cdot 6 = 12 = y + z.$

The requested difference is $\begin{aligned} x - y &= (x + z) - (y + z) \\ &= 16 - 12 = 4. \end{aligned}$

Thus, the correct answer is B.

2004 AMC 12A Problem 8

Answer: B

Difficulty rating: 1370

Solution:

Problem 8 in Other Years