GMAT Math : Right Triangles

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Example Question #5 : Dsq: Calculating An Angle In A Right Triangle

You are given two right triangles: $\triangle ABC$ with right angle , and $\triangle XYZ$ with right angle .

True or false: $\triangle ABC \sim \triangle XYZ$

Statement 1: $\angle A \cong \angle Y$

Statement 2: $\angle B \cong \angle X$

Possible Answers:

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 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

For $\triangle ABC \sim \triangle XYZ$ , it is necessary that corresponding angles be congruent - specifically, $m \angle A = m \angle X$ and $m \angle B = m \angle Y$ . We show that the statements together are insufficient by assuming them both to be true and examining two cases:

Case 1: $m \angle A = m \angle B = m \angle X = m \angle Y = 45^{\circ }$ .

Case 2: $m \angle A = m \angle Y = 30^{\circ }$ and $m \angle B = m \angle X= 60^{\circ }$ .

Both cases fit the main body of the problem and both statements, but in the first case, $m \angle A = m \angle X$ , and in the second case, $m \angle A \ne m \angle X$ . The statement $\triangle ABC \sim \triangle XYZ$ holds only in the first case but not in the second. (Note that in both cases, $\triangle ABC \sim \triangle YXZ$ , but this is a different statement.)

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Example Question #6 : Dsq: Calculating An Angle In A Right Triangle

Given a right triangle $\bigtriangleup ABC$ with right angle $\angle B$ , what is the measure of $\angle A$ ?

Statement 1: $\bigtriangleup ABC \sim \bigtriangleup CBA$

Statement 2: AB = BC

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Corresponding angles of similar triangles - including the same triangle, considered in different configurations - are congruent. It follows from Statement 1 alone that $\angle A \cong \angle C$ . The measures of the acute angles of a right triangle add up to $90^{\circ }$ , so:

$m \angle A + m \angle C =90 ^{\circ }$

$m \angle A + m \angle A =90 ^{\circ }$

$2 \cdot m \angle A =90 ^{\circ }$

$m \angle A =45 ^{\circ }$

Assume Statement 2 alone. By the 45-45-90 Theorem, since the legs of the right triangle are of equal length, the acute angles of the triangle, one of which is $\angle A$ , measures $45 ^{\circ }$ .

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Example Question #7 : Dsq: Calculating An Angle In A Right Triangle

Given a right triangle $\bigtriangleup ABC$ with right angle $\angle B$ , what is the measure of $\angle A$ ?

Statement 1: An arithmetic sequence can be written that begins as follows: $m \angle A, m \angle C, 75^{\circ }$

Statement 2: $m \angle A$ is a whole number and a multiple of both 5 and 7.

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone, and let be the measure of $\angle A$ . The sum of the measures of the acute angles of a right triangle, $\angle A$ and $\angle C$ , is $90^{\circ }$ , so

$m \angle A + m \angle C = 90^{\circ }$

$X + m \angle C = 90^{\circ }$

$X + m \angle C -X = 90^{\circ } -X$

$m \angle C = 90^{\circ } -X$

The first two terms of the arithmetic sequence given are and 90 - X . In an arithmetic sequence, each term is obtained by adding the same number, or common difference. This is the difference of the second and first terms, or

$90 ^{\circ }- X - X = 90^{\circ } - 2X$ ,

which is added to the second term to get the third term:

$(90^{\circ } - X )+ (90^{\circ } - 2X) = 180 ^{\circ }- 3X$

The third term is $75^{\circ }$ , so

$180 ^{\circ } - 3X = 75^{\circ }$

$180 ^{\circ } - 3X - 180 ^{\circ } = 75^{\circ } - 180 ^{\circ }$

$- 3X = - 105 ^{\circ }$

$- 3X \div (-3) = - 105 ^{\circ } \div (-3)$

$X = 35 ^{\circ }$ , the measure of $\angle A$ .

Assume Statement 2 alone. $\angle B$ is the right angle of the triangle, so $\angle A$ is an acute angle; it will have measure less than $90^{\circ }$ . Its measure is a whole number less than 90 which is, by Statement 2, a multiple of 5 and 7. However, both 35 and 70 are multiples of 5 and 7, so $m \angle A$ can be either, and no information is given that eliminates either choice.

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Example Question #8 : Dsq: Calculating An Angle In A Right Triangle

A right triangle $\bigtriangleup ABC$ with right angle $\angle B$ ; all of its interior angles have degree measures that are whole numbers. What is the measure of $\angle A$ ?

Statement 1: $m \angle A$ is a perfect cube integer.

Statement 2: $m \angle C$ is a multiple of both 7 and 9.

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

An acute angle must have measure less than $90^{\circ }$ ; both non-right angles of the triangle, $\angle A$ and $\angle C$ , must be acute, and thus have measure less than $90^{\circ }$ .

Assume Statement 1 alone. There are four perfect cubes less than 90 - 1, 8, 27, and 64 - so $\angle A$ can have any one of these four degree measures. There is no way to eliminate any one of these four.

Assume Statement 2 alone. The least common multiple of 7 and 9 is 63, so, since $m \angle C$ is a multiple of both 7 and 9, it must be a multiple of 63. The only multiple of 63 less than 90 is 63 itself, so $m \angle C = 63^{\circ }$ . The acute angles of a right angle have degree measures that total $90^{\circ }$ , so

$m \angle A + m \angle C = 90^{\circ }$

$m \angle A + 63^{\circ } = 90^{\circ }$

$m \angle A = 90^{\circ } - 63^{\circ } = 27^{\circ }$

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Example Question #9 : Dsq: Calculating An Angle In A Right Triangle

A right triangle $\bigtriangleup ABC$ with right angle $\angle B$ ; all of its interior angles have degree measures that are whole numbers. What is the measure of $\angle A$ ?

Statement 1: $m \angle A$ is a multiple of both 6 and 7.

Statement 2: $m \angle C$ is a multiple of 6 and 8.

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

An acute angle must have measure less than $90^{\circ }$ ; both non-right angles of the triangle, $\angle A$ and $\angle C$ , must be acute.

Assume Statement 1 alone. The measure of $\angle A$ must be a multiple of 6 and 7 that is less than 90; however, there are at least two such numbers, 42 and 84. With no way to eliminate either choice, Statement 1 alone provides insufficient information.

Statement 2 alone also provides insufficient information, for similar reasons. $\angle C$ can have measure 24 or 48, for example; the measure of $\angle A$ is 90 minus the measure of $\angle C$ , so if $m \angle C$ cannot be determined definitively, neither can $m \angle A$ .

Now, assume both statements to be true. 6 and 7 have LCM 42, so any multiple of both must also be a multiple of 42. There are only two multiples of 42 less than 90 - 42 and 84.

If $m \angle A = 42^{\circ }$ , then, since the measures of the angles of the acute angles of a right triangle total $90^{\circ }$ ,

$m \angle A+ m \angle C = 90^{\circ }$

$42 ^{\circ }+ m \angle C = 90^{\circ }$

$m \angle C = 90^{\circ } - 42 ^{\circ } = 48 ^{\circ }$

48 is a multiple of both 6 and 8, so this scenario is consistent with both statements.

If $m \angle A = 84^{\circ }$ , then, since the measures of the angles of the acute angles of a right triangle total $90^{\circ }$ ,

$m \angle A+ m \angle C = 90^{\circ }$

$84^{\circ }+ m \angle C = 90^{\circ }$

$m \angle C = 90^{\circ } - 84^{\circ } = 6^{\circ }$

6 is not a multiple of 8, so this scenario is inconsistent with Statement 2.

Therefore, it can be determined that $m \angle A = 42^{\circ }$ .

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Example Question #10 : Dsq: Calculating An Angle In A Right Triangle

Is $\bigtriangleup ABC$ an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: The measures of the angles of the triangle, when they are arranged in ascending order, form an arithmetic sequence.

Statement 2: $m \angle A = 30^{\circ }$

Possible Answers:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. Let be the measure of the angle of second-greatest measure. Since the measures form an arithmetic sequence, the three angles measure y - d, y, y+d for some common difference . Their sum is $180^{\circ }$ , so

$(y - d) + y + (y+d) = 180^{\circ }$

$3y= 180^{\circ }$

$y= 60^{\circ }$

However, since we do not know that common difference, we cannot determine the other angle measures, or whether the triangle is acute, right, or obtuse. For example, if d = 1 , the widest of the angles measures $61^{\circ }$ ; if d = 31 , the widest angle measures $91^{\circ }$ . In the former case, the triangle is acute, having all of its angles measure less than $90^{\circ }$ ; in the latter case, the triangle is obtuse, having an angle that measures greater than $90^{\circ }$ .

Statement 2 alone provides insufficient information; a 30-30-120 triangle and a 30-60-90 triangle both fit this condition as well as the conditions of the measures of the angles of a triangle, but the former is obtuse and the latter is right.

Now assume both statements to be true. Then, since one angle measures $60^{\circ }$ by Statement 1, and a second measures $30^{\circ }$ , the third measures $\left [180 - (30+60) \right ]^{\circ } = 90^{\circ }$ . This angle is a right angle, so $\bigtriangleup ABC$ is a right triangle.

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

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Given: $\overline{AE}$ is the height of $\bigtriangleup ABC$ .

What is the area of $\bigtriangleup ABE$ ?

(1) $\bigtriangleup ABC$ is a right triangle where $\angle A$ is a right angle.

(2) AB=4 , EC=1

Possible Answers:

Statement (2) is sufficient

Both statements are sufficient

Each statement alone is sufficient

Statement (1) is sufficient

Statement (1) and (2) taken together are not sufficient

Correct answer:

Statement (1) and (2) taken together are not sufficient

Explanation:

Here, we would need to know the length of at least two sides of the triangle ABC, provided it is a right triangle, in order to calculate the area of ABE.

Statement 1 is insufficient alone because it only tells us a property of the triangle and gives us no information about the lengths of the sides. Similarily, statement 2 alone is insufficient because we can't tell the area of triangle ABE from what is given. We would need the length of the height and the length of EB.

Statements 1 and 2 taken together are insufficient, because even though the height divides the triangle in two similar triangles, we can't find any ratio to calculate the length of the height AE. Therefore, Statements 1 and 2 taken together are insufficient.

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

Find the area of the right triangle.

Statement 1): The hypotenuse is $2\sqrt2$ .

Statement 2): Both legs have a side length of .

Possible Answers:

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

EACH statement ALONE is sufficient.

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

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

BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question.

Correct answer:

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

Explanation:

Statement 1) only provides the hypotenuse of the triangle, but it does not imply that both legs of the right triangle are congruent sides. Of the three interior angles, only the right angle is known with the other 2 unknown interior angles.

Statement 1) does not have enough information to solve for the area of the triangle.

Statement 2) provides the lengths of both legs. The formula $A=\frac{bh}{2}$ can be used to solve for the area of the triangle.

Statement 2) can be used by itself to solve for the area of the triangle.

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

The lobby of a building is in the shape of a right triangle. The shortest side of the room is meters long. Find the number of tiles needed to cover the floor.

I) Each tile covers square centimeters.

II) The longest side of the lobby is five less than three times the shortest side.

Possible Answers:

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

To find the number of tiles needed, we need to find the area of the lobby and the area of one tile.

I) Gives us the area of one tile.

II) Gives us the length of the hypotenuse of the lobby.

Use II) along with the info given in the question to find the last side (Pythagorean Theorem).

$H=3SS-5 \rightarrow H=3(14)-5$

H=37

SS^2+MS^2=H^2 , where SS is the short side, MS is the middle side, and H is the hypotenuse.

14^2+MS^2=37^2

$MS=\sqrt{37^2-14^2}=\sqrt{1173}=34.2$

From there you can find the area of the lobby.

$A=\frac{1}{2}b\cdot h \rightarrow A=\frac{1}{2}34.2\cdot 14=239.4m^2$

Use I) along with the area of the lobby to find the number of tiles needed.

10,000cm^2=1m^2

$239.4m^2\cdot\frac{10,000cm^2}{1m^2}=2394000cm^2$

$\frac{2394000cm^2}{20cm^2}=119700 \text{ tiles}$

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

What is the area of the right triangle?

The hypotenuse measures .
The height measures .

Possible Answers:

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.

Each statement alone is 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:

Neither statement provides us with sufficient information to answer the question but both statements taken together are sufficient to answer the question.

In order to find the area of the right triangle we need both the height and base. We can use the Pythagorean Theorem to solve for the base.

c^2=a^2+b^2

b^2=5^2-4^2

$b=\sqrt{25-16}=\sqrt{9}=3cm$

Now we can find the area:

$\frac{1}{2}(3)(4)=6cm^2$

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GMAT Math : Right Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Dsq: Calculating An Angle In A Right Triangle

Example Question #6 : Dsq: Calculating An Angle In A Right Triangle

Example Question #7 : Dsq: Calculating An Angle In A Right Triangle

Example Question #8 : Dsq: Calculating An Angle In A Right Triangle

Example Question #9 : Dsq: Calculating An Angle In A Right Triangle

Example Question #10 : Dsq: Calculating An Angle In A Right Triangle

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

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

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

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

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