In rectangle ABCD, AB = 12, and diagonal AC forms a 30^ angle with side AB. What is the area of rectangle ABCD?

In rectangle ABCD, AB = 12, and diagonal AC forms a 30^ angle... | SAT Right triangles and trigonometry Question, Answer Explained

Question bank / Right triangles and trigonometry

Math · Geometry and TrigonometryHard

In rectangle $ABCD$ , $AB = 12$ , and diagonal $\overline{AC}$ forms a $30^\circ$ angle with side $\overline{AB}$ . What is the area of rectangle $ABCD$ ?

$24\sqrt{3}$

$72$

$48\sqrt{3}$

$144\sqrt{3}$

Show answer and explanation

Answer: C

Triangle $ABC$ has a right angle at $B$ and a $30^\circ$ angle at $A$ . Side $\overline{BC}$ is opposite the $30^\circ$ angle and side $\overline{AB}$ is adjacent to it, so $BC = 12 \tan 30^\circ = 12 \cdot \frac{\sqrt{3}}{3} = 4\sqrt{3}$ . The area of the rectangle is $AB \times BC = 12 \times 4\sqrt{3} = 48\sqrt{3}$ .

Choice B uses $BC = 12 \sin 30^\circ = 6$ , which would be right only if $12$ were the length of the diagonal instead of side $\overline{AB}$ .

Why the other choices are wrong

A. 24 times the square root of 3 is the area of triangle ABC, half the rectangle.
B. This uses sine and gets BC = 6, which would be right only if 12 were the diagonal.
D. This uses tan 60 degrees instead of tan 30 degrees for side BC.

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