DSQ: Calculating the length of the side of a right triangle - GMAT Quantitative Reasoning

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Question

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

Statement 1:

Statement 2:

Answer

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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Question

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

Answer

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

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Question

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.

Answer

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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Question

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.

Answer

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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Question

What is the base length of the right triangle?

The width is four times the length.

The area of the right triangle is .

Answer

Statement 1: All we're given is the equation for finding the width, , 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.

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

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

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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Question

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

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

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

Answer

Either statement alone is sufficient.

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

Also, since $\angle B$ is the right angle, $\overline{AC}$ is the hypotenuse, and $\overline{BC}$ , opposite the $30 ^{\circ }$ 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 $\overline{AB}$ . From Statement 1 alone, $\overline{AB}$ has length $\frac{\sqrt{3}}{2}$ times that of the hypotenuse, or $\frac{\sqrt{3}}{2} \cdot 22 = 11\sqrt{3}$ . From Statement 2 alone, $\overline{AB}$ has length $\sqrt{3}$ of the shorter leg, or $11\sqrt{3}$ .

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Question

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

Statement 1: $\bigtriangleup ABC$ can be inscribed in a circle with circumference $20 \pi$ .

Statement 2: $\bigtriangleup ABC$ can be inscribed in a cricle with area $100 \pi$ .

Answer

From Statement 1 alone, the circumscribed circle has as its diameter the circumference $20 \pi$ divided by $\pi$ , or 20. From Statement 2 alone, the circle has area $100 \pi$ , so its radius can be found using the area formula;

$\pi r^{2}= A$

$\pi r^{2}= 100 \pi$

$r^{2}= 100$

The diameter is the radius doubled, which here is 20.

The hypotenuse of a right triangle is a diameter of the circle that circumscribes it, so the diameter of the circle gives us the length of the hypotenuse. However, we are looking for the length of a leg, $\overline{AB}$ . Either statement alone gives us only the length of the hypotenuse, which, without other information, does not give us any further information about the right triangle.

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Question

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

Statement 1: $\bigtriangleup ABC$ has area 24.

Statement 2: $\bigtriangleup ABC$ can be circumscribed by a circle with area $25 \pi$ .

Answer

Since $\angle B$ is given as the right angle of the triangle $\bigtriangleup ABC$ , we are being asked to evaluate the length of hypotenuse $\overline{AC}$ .

Statement 1 alone gives insufficient information. We note that the area of a right triangle is half the product of the lengths of its legs, and we examine two scenarios:

Case 1:

The area is $\frac{1}{2} \cdot 6 \cdot 8 = 24$

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

$\sqrt{6^{2}+8^{2}} = \sqrt{36+64} = \sqrt{100}= 10$

Case 2:

The area is $\frac{1}{2} \cdot 4 \cdot 12 = 24$

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

$\sqrt{4^{2}+12^{2}} = \sqrt{16+144} = \sqrt{160}= \sqrt{16} \cdot \sqrt{10} = 4\sqrt{10}$

Both triangles have area 24 but the hypotenuses have different lengths.

Assume Statement 2 alone. A circle that circumscribes a right triangle has the hypotenuse of the triangle as one of its diameters, so the length of the hypotenuse is the diameter - or, twice the radius - of the circle. Since the area of the circumsctibed circle is $25 \pi$ , its radius can be determined using the area formula:

$\pi r^{2}= A$

$\pi r^{2}= 25 \pi$

$r^{2}= 25$

The diameter - and the length of hypotenuse $\overline{AC}$ - is twice this, or 10.

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Question

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

Statement 1: and

Statement 2: $\bigtriangleup ABC$ is not a 30-60-90 triangle.

Answer

Statement 1 alone gives insufficient information. and , but it is not clear which of the three sides is the hypotenuse of $\bigtriangleup ABC$ . $\overline {BC}$ is not the longest side, so we know that $\overline {AB}$ or $\overline {AC}$ is the hypotenuse, and the other is the second leg. We explore the two possibilities:

If $\overline {AB}$ is the hypotenuse, then the legs are $\overline {AC}$ and $\overline {BC}$ ; since the lengths of the legs are 12 and 24, by the Pythagorean Theorem, $\overline {AB}$ has length

$\sqrt{24^{2}+12^{2}} = \sqrt{576+144}= \sqrt{720} = \sqrt{144} \cdot \sqrt{5} = 12 \sqrt{5}$ .

If $\overline {AB}$ is a leg, then the hypotenuse, being the longest side, is $\overline {AC}$ , and $\overline {BC}$ is the other leg; by the Pythagorean Theorem, $\overline {AB}$ has length

$\sqrt{24^{2}-12^{2}} = \sqrt{576-144}= \sqrt{432} = \sqrt{144} \cdot \sqrt{3} = 12 \sqrt{3}$ .

Statement 2 alone gives insufficient information in that it only gives information about the angles, not the sides.

Assume both statements are true. If $\overline {AC}$ is the hypotenuse and $\overline {BC}$ is a leg, then, since the hypotenuse measures twice the length of a leg from Statement 1, the triangle is 30-60-90, contradicting Statement 2. Therefore, by elimination, $\overline {AB}$ is the hypotenuse, and, consequently, $AB= 12 \sqrt{5}$ .

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Question

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

Statement 1: and

Statement 2: $\bigtriangleup ABC$ has a $60^{\circ }$ angle.

Answer

Assume Statement 1 alone. Since we do not know whether $\overline {AB}$ is the hypotenuse or a leg of $\bigtriangleup ABC$ , we can show that can take one of two different values.

Case 1: If $\overline {AB}$ is the hypotenuse, then the legs are $\overline {AC}$ and $\overline {BC}$ ; since their lengths are 10 and 20, by the Pythagorean Theorem, $\overline {AB}$ has length

$\sqrt{20^{2}+10^{2}} = \sqrt{400+100}= \sqrt{500} = \sqrt{100} \cdot \sqrt{5} = 10 \sqrt{5}$ .

Case 2: If $\overline {AB}$ is a leg, then the hypotenuse, being the longest side, is $\overline {AC}$ , and $\overline {BC}$ is the other leg; by the Pythagorean Theorem, $\overline {AB}$ has length

$\sqrt{20^{2}-10^{2}} = \sqrt{400-100}= \sqrt{300}= \sqrt{100} \cdot \sqrt{3} = 10 \sqrt{3}$ .

Statement 2 gives insufficient information, since it only clues us in to the measures of the angles, not the sides.

Now assume both statements. Since one of the angles of the right triangle has measure $60^{\circ }$ , the other has measure $90^{\circ } - 60^{\circ }= 30^{\circ }$ ; the triangle is a 30-60-90 triangle, and therefore, its hypotenuse has twice the length of its shorter leg. Since, from Statement 1, $BC \cdot 2 = AC$ , $\overline {AC}$ is the hypotenuse, and $\overline {AB}$ is the longer leg, the length of which, by the 30-60-90 Theorem, is $\sqrt{3}$ times that of shorter leg $\overline {BC}$ , or $10\sqrt{3}$ .

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Question

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

Statement 1:

Statement 2:

Answer

The hypotenuse of a right triangle is its longest side. From Statement 1 alone, we can eliminate only $\overline{AC}$ as the hypotenuse, and from Statement 2 alone, we can eliminate only $\overline{BC}$ . From Statements 1 and 2 together, we can eliminate both, leaving $\overline{AB}$ as the hypotenuse.

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Question

Given $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , which, if either, is the longer of the two: $\overline{AC}$ or $\overline{DF}$ ?

Statement 1:

Statement 2: $\angle B$ is a right angle and $\angle E$ is an obtuse angle

Answer

Statement 1 alone gives insufficient information, as it only establishes congruence between two pairs of corresponding sides; without a third congruence or noncongruence between sides or angles, it cannot be established whether the triangles themselves are congruent. Statement 2 alone is insufficient, since it only compares two angles without giving any information about sidelengths, whether absolute or relative.

Assume both statements to be true. From Statement 1, we have the congruence statements $\overline{AB }\cong \overline{DE }$ and $\overline{BC }\cong \overline{EF}$ , and from Statement 2, we have that the included angle $\angle B$ from $\bigtriangleup ABC$ has measure $90^{\circ }$ , and that the included angle $\angle E$ from $\bigtriangleup DEF$ , being obtuse, has, by definition, measure greater than this. This sets up the conditions of the Side-Angle-Side Inequality Theorem, or Hinge Theorem, which states that in this situation, the third side opposite the greater angle is longer than the third side opposite the lesser. Therefore, it can be deduced that .

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Question

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

Statement 1:

Statement 2: $\angle A \cong \angle C \cong \angle D \cong \angle F$

Answer

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.

Assume Statement 1 alone. We have that , , and $\angle B \cong \angle E$ , both being right angles, thereby establishing congruence between two pairs of sides and a pair of included angles. By the Side-Angle-Side Theorem, $\bigtriangleup ABC \cong \bigtriangleup DEF$ , and, consequently $\overline{AC}$ and $\overline{DF}$ have equal length.

Assume Statement 2 alone. We are only given information about the angle measures, but nothing about the lengths of the sides - actual lengths or comparisons. We can make no conclusions about which hypotenuse is longer.

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Question

$\bigtriangleup ABC$ 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: $m \angle D = 45^{\circ }$

Answer

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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Question

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

Statement 1:

Statement 2:

Answer

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:

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:

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, and , so . 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 and , by the subtraction property of inequality,

and

It follows from and 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

,

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

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Question

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

Statement 1:

Statement 2:

Answer

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, . Since, from Statement 1, , it follows that .

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

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Question

Refer to the above figure. What is the length of $\overline{AC}$ ?

Statement 1:

Statement 2:

Answer

As each statement alone only gives the length of one segment, neither statement alone is sufficient to find the length of any other segment, including, in particular, $\overline{AC}$ .

Assume both statements to be true. can be found by way of the Pythagorean Theorem:

$AD = \sqrt{(AB)^{2}-(BD)^{2}}$

$= \sqrt{12^{2}-10^{2}}$

$= \sqrt{144-100}$

$= \sqrt{44}$

$= \sqrt{4} \cdot \sqrt{11}$

$=2\sqrt{11}$

Now note that $\overline{BD}$ is the altitude of the right triangle from the vertex of the right angle, which divides the right triangle into two triangles similar to each other and the large triangle. Specifically,

$\bigtriangleup ABC \sim \bigtriangleup ADB$

By similarity,

$\frac{AC}{AB}= \frac{AB}{AD}$

$\frac{AC}{AB}\cdot AB= \frac{AB}{AD} \cdot AB$

$AC = \frac{\left (AB \right )^{2}}{AD}$

$AC = \frac{12^{2}}{4 \sqrt{11}} = \frac{144}{4 \sqrt{11}} = \frac{36}{ \sqrt{11}}$ .

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Question

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

Statement 1: $\overline{AB } \cong \overline{AC}$

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

Answer

Assume Statement 1 alone. The hypotenuse of a right triangle is longer than either of its two other sides; since $\overline{AB}$ and $\overline{AC}$ are equal in length, neither is the hypotenuse. This leaves $\overline{BC}$ as the hypotenuse.

Assume Statement 2 alone. Since $\angle B \cong \angle C$ , neither can be the right angle. Therefore, $\angle A$ is the right angle, and its opposite side, $\overline{BC}$ , is the hypotenuse.

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Question

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: $m \angle A > m \angle B$

Answer

Since we are comparing angles, we need to identify the angle of greatest measure; in a right triangle, the angle of greatest measure is the right angle, and the side opposite it is the hypotenuse.

Statement 1 is insufficient, since we can eliminate only angle $\angle C$ as the right angle, and, subsequently, only $\overline{AB}$ as the hypotenuse. Similarly, Statement 2 is insufficent, since we can eliminate only angle $\angle B$ as the right angle, and, subsequently, only $\overline{AC}$ as the hypotenuse. But if we are given both statements, we can eliminate $\overline{AB}$ and $\overline{AC}$ as the hypotenuse, leaving $\overline{BC}$ as the hypotenuse.

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Question

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

Answer

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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