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

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Example Question #1 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Data sufficiency question- do not actually solve the question

The lengths of two sides of a triangle are 6 and 8. What is the length of the third side?

1. The length of the longest side is 8.

2. The triangle contains a right angle.

Possible Answers:

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

Each statement alone is sufficient to answer the question

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Explanation:

Knowing that the triangle has a right angle indicates the question can be solved using the Pythagorean Theorem, however it is unclear which side is the hypotenuse. For example, if 8 is not the hypotenuse, the length of the third side is 10. If 8 is the hypotenuse, the length of the third side is 5.3.

Additionally, if you only know that 8 is the longest side, the length of the third side could be anything greater than 2 and less than 8. Therefore, having both pieces of data will allow you to solve the problem.

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Example Question #2 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Which of the three sides of $\Delta ABC$ has the greatest measure?

Statement 1: $\angle A$ and $\angle B$ are complementary angles.

Statement 2: $\angle C$ is not an acute angle.

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

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

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The side opposite the angle of greatest measure is the longest of the three, so if we can determine which angle is the longest, we can answer this question.

It follows from Statement 1 by definition that $m\angle A + m\angle B = 90^{\circ }$ , so, since the measures of the three angles total $180^{\circ }$ , $m\angle C = 90^{\circ }$ , making $\angle C$ right and the other two acute. This proves $\angle C$ has the greatest measure of the three.

It follows from Statement 2 that $\angle C$ is either right or obtuse; therefore, $m\angle C \geq 90^{\circ }$ . Subsequently, the other two angles are acute, so again, $\angle C$ has the greatest measure of the three.

From either statement alone, we can therefore identify $\overline{AB}$ as the side of greatest measure.

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Example Question #3 : Dsq: Calculating The Length Of The Side Of A Right Triangle

What is the base length of the right triangle?

The width is four times the length.
The area of the right triangle is .

Possible Answers:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Each statement alone is sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Explanation:

Statement 1: All we're given is the equation for finding the width, w=4l , which we'll use in the next statement.

Statement 2: Using the information from statement 1, we can set up an equation and solve for the length.

$A=\frac{1}{2}l\times w= \frac{1}{2} (l)(4l)=\frac{1}{2}4l^2$

$18=\frac{1}{2}\times 4 l^2$

18=2l^2

9=l^2

l=3cm

Statement 2 alone would not have provided sufficient information because we would have ended up with

$36=l \times w$

and would not have been able to determine what the the values were.

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Example Question #4 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Given $\bigtriangleup ABC$ is a right triangle, which side is the hypotenuse - $\overline{AB}$ , $\overline{AC}$ , or $\overline{BC}$ ?

Statement 1: AC > BC

Statement 2: AB > BC

Possible Answers:

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

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

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

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The hypotenuse of a right triangle is its longest side.

Assume both statements are true. Then we know that $\overline{BC}$ , being shorter than either of the other sides, cannot be the hypotenuse, but without further information, we cannot tell which of the two other sides is longer than the other. Therefore, we cannot identify the hypotenuse for certain.

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Example Question #5 : Dsq: Calculating The Length Of The Side Of A Right Triangle

$\bigtriangleup ABC$ has right angle $\angle B$ ; $\bigtriangleup DEF$ has right angle $\angle E$ . Which, if either, is longer, $\overline{AC}$ or $\overline{DF}$ ?

Statement 1: AC = DE

Statement 2: AC > EF

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

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

BOTH statements TOGETHER are insufficient to answer the question.

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

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

Correct answer:

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

Explanation:

Assume Statement 1 alone. Since $\angle B$ and $\angle E$ are the right angles of their respective triangles, $\overline{AC}$ and $\overline{DF}$ , the segments opposite the right angles, are their hypotenuses, and, subsequently, their longest sides. Specifically, DF > DE . Since, from Statement 1, AC = DE , it follows that DF > AC .

Assume Statement 2 alone. Again, $\overline{DF}$ is the longest side of its triangle, so DF > EF . But we cannot determine whether AC> DF > EF or DF >AC> EF without further information.

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Example Question #6 : Dsq: Calculating The Length Of The Side Of A Right Triangle

$\bigtriangleup ABC$ is a right triangle. Evaluate .

Statement 1: AC = 25

Statement 2: BC = 25

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

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

Explanation:

Neither statement alone gives enough information to find , as each alone gives only one sidelength.

Assume both statements are true. While neither side is indicated to be a leg or the hypotenuse, the hypotenuse of a right triangle is longer than either leg; therefore, since $\overline {AC}$ and $\overline {BC}$ are of equal length, they are the legs. $\overline {AB}$ is the hypotenuse of an isosceles right triangle with legs of length 10, and by the 45-45-90 Theorem, the length of $\overline{AB}$ is $\sqrt{2}$ times that of a leg, or $10 \sqrt{2}$ .

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Example Question #1 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Given $\bigtriangleup ABC$ is a right triangle, which side is the hypotenuse - $\overline{AB}$ , $\overline{AC}$ , or $\overline{BC}$ ?

Statement 1: $m \angle B = 60^{\circ }$

Statement 2: $\frac{AB}{BC} = 2$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

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

BOTH statements TOGETHER are insufficient to answer the question.

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

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

Correct answer:

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

Explanation:

The side opposite of the right angle is the hypotenuse. Statement 1 alone eliminates $\angle B$ as the right angle, and, as a consequence, $\overline{AC}$ as the hypotenuse - but only $\overline{AC}$ .

From Statement 2 alone, we have that $\frac{AB}{BC} = 2$ , meaning that

$\frac{AB}{BC} \cdot BC= 2 \cdot BC$

and

$AB= 2 \cdot BC$

Since $\overline{BC}$ is shorter than $\overline{AB}$ , $\overline{BC}$ , and only $\overline{BC}$ , is eliminated as the hypotenuse.

If both statements are assumed to be true, however, then both $\overline{AC}$ and $\overline{BC}$ can be eliminated as the hypotenuse, leaving $\overline{AB}$ as the only choice.

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Example Question #8 : Dsq: Calculating The Length Of The Side Of A Right Triangle

$\bigtriangleup ABC$ has right angle $\angle B$ ; $\bigtriangleup DEF$ has right angle $\angle E$ . Which, if either, is longer, $\overline{AC}$ or $\overline{DF}$ ?

Statement 1: AB + BC < DE+EF

Statement 2: AB = DE

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

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

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

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

We are being asked to compare the lengths of the hypotenuses of the two triangles, since $\overline{AC}$ and $\overline{DF}$ are the sides opposite the right angles of their respective triangles.

Statement 1 alone gives insufficient information, as shown by examining these two cases.

Case 1: AB =BC = 7, DE=EF = 8

By the Pythagorean Theorem, the hypotenuse $\overline{AC}$ has length

$\sqrt{7^{2}+7^{2}} = \sqrt{49+49} = \sqrt{98}$

The hypotenuse $\overline{DF}$ has length

$\sqrt{8^{2}+8^{2}} = \sqrt{64+64} = \sqrt{128}$

Case 2: AB =1, BC = 13, DE=EF = 8

The hypotenuse $\overline{AC}$ has length

$\sqrt{1^{2}+13^{2}} = \sqrt{1+ 169} = \sqrt{170}$

and, as in Case 1, $\overline{DF}$ has length $\sqrt{128}$ .

In both cases, AB + BC = 14 and DE+EF= 16 , so AB + BC < DE+EF . But in the first case, $\overline{DF}$ was longer than $\overline{AC}$ , and in the second case, the reverse was true.

Statement 2 is insufficient in that it only gives us the congruence of one set of corresponding legs; without further information, it is impossible to determine which hypotenuse is longer.

Now assume both statements are true. Since AB + BC < DE+EF and AB = DE , by the subtraction property of inequality,

AB + BC -AB < DE+EF - DE

and

BC < EF

It follows from AB = DE and BC < EF that $\left (AB \right )^{2} =\left ( DE \right )^{2}$ and $\left (BC \right )^{2} < \left (EF \right )^{2}$ ; by the addition property of inequality,

$\left (AB \right )^{2} + \left (BC \right )^{2} <\left ( DE \right )^{2}+ \left (EF \right )^{2}$

By the Pythagorean Theorem,

$\left (AB \right )^{2} + \left (BC \right )^{2} = (AC)^{2}$

and

$\left ( DE \right )^{2}+ \left (EF \right )^{2} = \left ( DF\right )^{2}$ ,

so the above inequality becomes, by substitution,

$\left (A C \right )^{2} <\left ( D F \right )^{2}$

and

A C < D F ,

proving that $\overline{DF}$ is longer than $\overline{AC}$ .

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Example Question #9 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Given $\bigtriangleup ABC$ is a right triangle, which side is the hypotenuse - $\overline{AB}$ , $\overline{AC}$ , or $\overline{BC}$ ?

Statement 1: $m \angle B > m \angle C$

Statement 2: $m \angle A > m \angle C$

Possible Answers:

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

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

In a right triangle, the angle of greatest measure is the right angle, and the side opposite it is the hypotenuse.

Assume both statements are true. We can eliminate $\angle C$ as the right angle, as it has measure less than both $\angle A$ and $\angle B$ . However, we have no information that tells us which of $\angle A$ and $\angle B$ has the greater measure, so we cannot determine which is the right angle. Subsequently, we cannot eliminate either of their opposite sides, $\overline{BC}$ or $\overline{AC}$ , respectively, as the hypotenuse.

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Example Question #10 : Dsq: Calculating The Length Of The Side Of A Right Triangle

$\bigtriangleup ABC$ has right angle $\angle B$ ; $\bigtriangleup DEF$ has right angle $\angle E$ . Which, if either, is longer, $\overline{AC}$ or $\overline{DF}$ ?

Statement 1: $m \angle A = 30^{\circ }$

Statement 2: $m \angle D = 45^{\circ }$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

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

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

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The two statements together only give information about the angle measures of the two triangles. Without any information about the relative or absolute lengths of the sides, no comparison can be drawn between their hypotenuses.

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

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Example Question #1 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Example Question #2 : Dsq: Calculating The Length Of The Side Of A Right Triangle

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