Right Triangles - GMAT Quantitative Reasoning

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Question

Which interior angle of $\Delta ABC$ has the greatest measure?

Statement 1: $\left (AB \right )^{2} + \left (BC \right )^{2} =\left (AC \right )^{2}$

Statement 2: $\angle B$ is a right angle.

Answer

If Statement 1 is assumed, then by the converse of the Pythagorean Theorem, the triangle is a right triangle with right angle $\angle B$ , which is explicitly stated in Statement 2. If $\angle B$ is a right angle, then the other two angles are acute, since a triangle must have at least two acute angles. A right angle measures $90 ^{\circ }$ and an acute angle measures less, so from either statement, we can deduce that $\angle B$ is the angle with greatest measure.

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Question

Note: Figure NOT drawn to scale.

$\angle A,\angle B$ are acute. Is $\angle C$ a right angle?

Statement 1: $m \angle A + m \angle B <90 ^{\circ }$

Statement 2:

Answer

A right triangle must have its two acute angles complementary; if Statement 1 is assumed, then this is false, the triangle is not a right triangle, and $\angle C$ is not a right angle.

If Statement 2 is assumed, then we apply the converse of the Pythagorean Theorem to show that the triangle is not right. The sides of a triangle have the relationship

$c^{2} = a^{2} + b^{2}$

only in a right triangle. If , then the statement to be tested would be

$260^{2} = 70^{2} + 240^{2}$

This statement is false, so the triangle is not a right triangle, and $\angle C$ is not a right angle.

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Question

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: $\angle B \cong \angle X$

Answer

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, $m \angle A e m \angle X$ . 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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Question

Given: $\bigtriangleup ABC$ is a right triangle with height $\overline{AD}$ and $\angle A$ is a right angle.

What is the size of $\angle {ABC}$ ?

(1) $\overline{AD}=\overline{DC}$

(2) $\overline{AC}=4$

Answer

In order to find the angles of right triangle ABC, we would need to find the length of the sides and maybe found that the triangle is isoceles, or is a special triangle with angles 30-60-90.

Statement one tells us that the height is equal to half the hypothenuse of the triangle. From that we can see that the triangle is isoceles. Indeed, an isoceles right triangle will always have its height equal to half the length of the hypothenuse. Therefore we will know that both angles are 45 degrees. Statement 1 alone is sufficient.

Statement 2 alone is insufficient because we don't know anything about the other sides of the triangle. Therefore it doesn't help us.

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Question

$\bigtriangleup ABC$ is a right triangle , where $\angle A$ is a right angle, and $\overline{AE}$ is a height of the triangle. What is the measurement of $\angle ABC$ ?

(1) $\angle CAE=30^{\circ}$

(2) $\angle EAB=60^{\circ}$

Answer

Since we are already told that triangle ABC is a right triangle, we just need to find information about other angles or other sides.

Statement 1 allows us to calculate $\angle ACE$ , simply by using the sum of the angles of a triangle, since we know AEC is also a right triangle because AE is the height.

Statement 2 is also sufficient because it allows us to know angle $\angle ACB$ . Indeed, in a right triangle, the height divides the triangles in two triangles with similar properties. Therefore angle $\angle ACB$ is the same as $\angle EAB$ .

Therefore, each statement alone is sufficient.

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Question

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

Answer

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 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 , the widest of the angles measures $61^{\circ }$ ; if , 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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Question

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:

Answer

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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Question

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.

Answer

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 . 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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Question

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.

Answer

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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Question

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.

Answer

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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Question

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) ,

Answer

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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Question

Find the area of the right triangle.

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

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

Answer

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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Question

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.

Answer

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).

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

, where SS is the short side, MS is the middle side, and H is the hypotenuse.

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

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.

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

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

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Question

What is the area of the right triangle?

The hypotenuse measures .

The height measures .

Answer

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.

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

Now we can find the area:

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

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Question

What is the length of the height $\overline{AE}$ of right triangle , where $\angle A$ is a right angle?

(1)

(2)

Answer

Since we are told that triangle ABC is a right triangle, to find the height, we just need the length of at least 2 other sides. From there, we can find the length of the height since in a right triangle, the height divides the triangle into two triangles with the same proportions. In other words $\frac{AE}{AC}=\frac{AB}{BC}$ . Therefore, we need to know the length of the sides of the triangle.

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Question

$\bigtriangleup ABC$ is a right triangle where $\angle A$ is a right angle. What is the length of the height $\overline{AD}$ ?

(1)

(1) $\angle ACB=45^{\circ}$

Answer

To know the length of the height triangle, we would need to know the lengths of the triangle or the angles to have more information about the triangle.

Statement 1 only gives us a length of a side. There is nothing more we can calculate from what we know so far.

Statement 2 alone tells us that the triangle is isoceles. Indeed, ABC is a right triangle, if one of its angle is 45 degrees, than so must be another. Now, we are able to tell that the length of the height would be the same as half the hypothenuse. A single side would be sufficient to answer the problem. Statment 1 gives us that information. Therefore, both statements together are sufficient.

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Question

Consider right $\Delta JKL$ .

I) The longest side, , has a length of meters.

II) ${}\angle K= \angle L= \frac{\pi}{4}$ .

What is the height of $\Delta JKL$ ?

Answer

The height of a right triangle will be one of its side lengths.

I) tells us the length of our hypotenuse.

II) gives us the other two angle measurements.

They are both 45 degrees, which makes JKL a 45/45/90 triangle with side length ratios of $1/1/\sqrt2$ .

Which we can use to find the height.

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Question

What is the height of the right triangle?

The area of the right triangle is .

The base of the right triangle measures .

Answer

Statement 1:

$A=\frac{1}{2}\times b\times h$

$10=\frac{1}{2} \times b \times h$

$20=b \times h$

More information is required to answer the question because our base and height can be and or and

Statement 2: We're given the base so we can narrow down the information from Statement 1 to and . If the base is , then the height must be .

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

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Question

What is the height of the rigth triangle?

The area of the right triangle is .

The perimeter of the right triangle is $11+\sqrt{73} in$ .

Answer

Statement 1:

$A=\frac{1}{2} \times b \times h =12$

$b \times h =24$

Additional information is required because our base and height can be and , and , or and .

Statement 2:

$P=h + b + \sqrt{h^2 + b^2}=11+\sqrt{73}$

Even if we solve for our two values, we will not be able to determine which is the base and which is the height.

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

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Question

Two ships left New York at the same time. One ship has been moving due east the entire time at a speed of 50 nautical miles per hour. How far apart are the ships now?

The other ship has been moving due south the entire time.

The other ship has been moving at a rate of 60 nautical miles per hour the entire time.

Answer

In order to determine how far apart the ships are now, it is necessary to know how far each ship is from New York now. Even knowing both statements, however, you only know that the paths are at right angles, and that the ratio of the two distances is 6 to 5. Without knowing the time elapsed, which is not given, you cannot tell how far apart the ships are.

The answer is that both statements together are insufficent to answer the question.

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