GMAT Math : DSQ: Calculating the length of the side of a right triangle

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Example Question #21 : Dsq: Calculating The Length Of The Side Of A Right Triangle

$\bigtriangleup ABC$ $\displaystyle \bigtriangleup ABC$ is a right triangle with right angle $\angle B$ $\displaystyle \angle B$. Evaluate $\displaystyle AB$.

Statement 1: AC = 22 $\displaystyle AC = 22$ and $m \angle A = 30^{\circ }$ $\displaystyle m \angle A = 30^{\circ }$.

Statement 2: BC = 11 $\displaystyle BC = 11$ and $m \angle C = 60^{\circ }$ $\displaystyle m \angle C = 60^{\circ }$.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Either statement alone is sufficient.

From either statement alone, it can be determined that $m \angle A = 30^{\circ }$ $\displaystyle m \angle A = 30^{\circ }$ and $m \angle C = 60^{\circ }$ $\displaystyle m \angle C = 60^{\circ }$; each statement gives one angle measure, and the other can be calculated by subtracting the first from $90^{\circ }$ $\displaystyle 90^{\circ }$, since the acute angles of a right triangle are complementary.

Also, since $\angle B$ $\displaystyle \angle B$ is the right angle, $\overline{AC}$ $\displaystyle \overline{AC}$ is the hypotenuse, and $\overline{BC}$ $\displaystyle \overline{BC}$, opposite the $30 ^{\circ }$ $\displaystyle 30 ^{\circ }$ angle, the shorter leg of a 30-60-90 triangle. From either statement alone, the 30-60-90 Theorem can be used to find the length of longer leg $\overline{AB}$ $\displaystyle \overline{AB}$. From Statement 1 alone, $\overline{AB}$ $\displaystyle \overline{AB}$ has length $\frac{\sqrt{3}}{2}$ $\displaystyle \frac{\sqrt{3}}{2}$ times that of the hypotenuse, or $\frac{\sqrt{3}}{2} \cdot 22 = 11\sqrt{3}$ $\displaystyle \frac{\sqrt{3}}{2} \cdot 22 = 11\sqrt{3}$. From Statement 2 alone, $\overline{AB}$ $\displaystyle \overline{AB}$ has length $\sqrt{3}$ $\displaystyle \sqrt{3}$ of the shorter leg, or $11\sqrt{3}$ $\displaystyle 11\sqrt{3}$.

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Example Question #141 : Triangles

The longest side of a right triangle has a length of $13\textup{ in}$ $\displaystyle 13\textup{ in}$. If the base of the triangle is $5\textup{ in}$ $\displaystyle 5\textup{ in}$ long, how long is the other side of the triangle?

Possible Answers:

$12\textup{in}$ $\displaystyle 12\textup{in}$

$9\textup{in}$ $\displaystyle 9\textup{in}$

$11\textup{in}$ $\displaystyle 11\textup{in}$

$8\textup{in}$ $\displaystyle 8\textup{in}$

$10\textup{in}$ $\displaystyle 10\textup{in}$

Correct answer:

$12\textup{in}$ $\displaystyle 12\textup{in}$

Explanation:

This is a Pythagorean theorem question. The lengths of a right triangle are related by the following equation: $a^{2}+b^{2}=c^{2}.$ $\displaystyle a^{2}+b^{2}=c^{2}.$ In the problem statement, $c=13\textup{in}$ $\displaystyle c=13\textup{in}$ and $a=5\textup{in.}$ $\displaystyle a=5\textup{in.}$ Therefore, $b^{2}= c^{2}-a^{2}= 13^{2}-5^{2}=169-25=144. b=\sqrt{144}=12\textup{in.}$ $\displaystyle b^{2}= c^{2}-a^{2}= 13^{2}-5^{2}=169-25=144. b=\sqrt{144}=12\textup{in.}$

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GMAT Math : DSQ: Calculating the length of the side of a right triangle

Study concepts, example questions & explanations for GMAT Math

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Example Question #21 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Example Question #141 : Triangles

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