In triangle ABC, the measure of angle A is 45^ and the measure of angle B is 30^. The altitude from C to AB has length 6. What is the length of AB?

SAT Right triangles and trigonometry Practice Question 9 (Hard)

Question bank / Right triangles and trigonometry

Math · Geometry and TrigonometryHard

In triangle $ABC$ , the measure of angle $A$ is $45^\circ$ and the measure of angle $B$ is $30^\circ$ . The altitude from $C$ to $\overline{AB}$ has length $6$ . What is the length of $\overline{AB}$ ?

$6 + 6\sqrt{3}$

$12$

$12 + 6\sqrt{2}$

$6 + 2\sqrt{3}$

Show answer and explanation

Answer: A

Let $D$ be the foot of the altitude on $\overline{AB}$ . Triangle $ADC$ is a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle, so $AD = CD = 6$ . Triangle $BDC$ is a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle with $CD = 6$ opposite the $30^\circ$ angle, so $BD = 6\sqrt{3}$ . Thus $AB = 6 + 6\sqrt{3}$ .

Choice D uses $BD = \frac{6}{\sqrt{3}} = 2\sqrt{3}$ , which swaps the roles of the legs in the $30^\circ$ - $60^\circ$ - $90^\circ$ triangle.

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