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GRADUATE RECORD EXAMINATIONS®

Official G R E Quantitative Reasoning

Practice Questions, Volume 1

Chapter 5 – Geometry

Answer Key with Answers and Explanations

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Geometry

This document begins with the answer key for questions found in the Chapter 5 Geometry Practice Questions document. Following the answer key are the complete explanations for each question. If you wish to work through the questions before consulting the answers and explanations, please use the Chapter 5 Geometry Practice Questions document.

Answer Key

Question 1.

Answer: Choice C. The two quantities are equal.

Question 2.

Answer: Choice C. The two quantities are equal.

Question 3.

Answer: Choice A. Quantity A is greater.

Question 4.

Answer: Choice D. The relationship cannot be determined from the information given.

Question 5.

Answer: Choice B. Quantity B is greater.

Question 6.

Answer: Choice B. 500

Question 7.

Answer: Choice D. [pic] 18 times the square root of 3

Question 8.

Answer: Choice D. [pic] 6 pi

Question 9.

Answer: Choice B. [pic] 2 pi, minus 4

Question 10.

Answer: Choice E. 16 to 9

Question 11.

The answer to question 11 consists of four of the answer choices.

Choice B. 30

Choice C. 40

Choice E. 60

Choice G. 80

Question 12.

The answer to question 12 consists of three of the answer choices.

Choice A. Quadrant [pic] 1

Choice C. Quadrant [pic] 3

Choice D. Quadrant [pic] 4

Question 13.

In question 13 you were asked to enter an integer of a decimal. The answer to question 13 is 112.5.

Answers and Explanations

Question 1.

In the x y-plane, one of the vertices of square S is the point [pic] with coordinates 2 comma 2. The diagonals of S intersect at the point [pic] with coordinates 6 comma 6.

Quantity A: The area of S

Quantity B: 64

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.

Explanation for Question 1.

Since the point [pic] with coordinates 2 comma 2 is a vertex of square S and the point [pic] with coordinates 6 comma 6 is the midpoint of the diagonals, it follows that the point [pic] with coordinates 10 comma 10 is also a vertex of the square. Using this information you can sketch square S in the x y plane, labeling the point [pic] with coordinates 2 comma 2, the point [pic] with coordinates 6 comma 6, and the point [pic] with coordinates 10 comma 10, as shown in the following figure.

[pic]

Figure for Explanation for Geometry Question 1

Begin skippable part of figure description.

In the figure, the numbers 2, 6, and 10 appear along the x-axis and along the y-axis. The square lies above the x-axis and to the right of the y-axis, and its sides are parallel to the x- and y-axes, respectively. The point [pic] with coordinates 2 comma 2, is the lower-left vertex of the square and the point [pic] with coordinates 10 comma 10 is the upper-right vertex. The lower-right vertex is the point [pic] with coordinates 10 comma 2, and the upper-left vertex is the point [pic] with coordinates 2 comma 10.

The intersection of the diagonals of the square is at the point [pic] with coordinates 6 comma 6.

End skippable part of figure description.

From the figure, you can see that S has sides of length 8. Therefore Quantity A, the area of S, is [pic] 8 squared, or 64. Hence Quantity A is equal to Quantity B, 64, and the correct answer is Choice C.

This explanation uses the following strategies.

Strategy 2: Translate from Words to a Figure or Diagram

Strategy 4: Translate from a Figure to an Arithmetic or Algebraic Representation

Question 2.

Quantity A: The length of a side of a regular pentagon with a perimeter of 12.5

Quantity B: The length of a side of a regular hexagon with a perimeter of 15

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.

Explanation for Question 2.

A regular pentagon has 5 sides of equal length, so the length of a side of a regular pentagon is [pic] one fifth of its perimeter. Thus Quantity A, the length of a side of a regular pentagon with a perimeter of 12.5, is [pic] the fraction 12.5 over 5, or 2.5. A regular hexagon has 6 sides of equal length, so the length of a side of a regular hexagon is [pic] one sixth of its perimeter. Thus Quantity B, the length of a side of a regular hexagon with a perimeter of 15, is [pic] the fraction 15 over 6, or 2.5. So Quantity A and Quantity B are both equal to 2.5, and the correct answer is Choice C.

This explanation uses the following strategy.

Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

Question 3.

It is given that a line in the x y-plane contains the point [pic] with coordinates 5 comma 4 and the point [pic] with coordinates 2 comma negative one.

Quantity A: The slope of the line

Quantity B: 0

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.

Explanation for Question 3.

You can begin by sketching the line in the x y-plane and labeling the point [pic] with coordinates 5 comma 4 and the point [pic] with coordinates 2 comma negative one on the line, as shown in the figure.

[pic]

Figure for Explanation for Geometry Question 3

Begin skippable part of figure description.

In the figure, the point with coordinates [pic] 2 comma negative 1 is 2 units to the right of the y-axis and one unit below the x-axis, and the point with coordinates [pic] 5 comma 4 is 5 units to the right of the y-axis and 4 units above the x-axis. The line passing through these two points slants upward and to the right.

End skippable part of figure description.

From the figure, you can see that the line through the two points slants upward and to the right. So Quantity A, the slope of the line, is greater than 0. Since Quantity B is 0, the correct answer is Choice A. (Note that it is not necessary to calculate the slope of the line.)

This explanation uses the following strategy.

Strategy 2: Translate from Words to a Figure or Diagram

Question 4.

Refer to the figure.

[pic]

Figure for Geometry Question 4

The figure shows a triangle.

Begin skippable part of figure description.

The triangle appears to be a right triangle, with one leg of length 5 + y, one leg of length [pic] 12 minus y, and hypotenuse of length 13. The angle that appears to be a right angle is labeled [pic] x degrees.

End skippable part of figure description.

Quantity A: x

Quantity B: 90

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.

Explanation for Question 4.

The figure accompanying the question looks like a right triangle with legs of length 5 + y and [pic] 12 minus y and hypotenuse of length 13. If y = 0, then the sides of the triangle have lengths 5, 12, and 13. This triangle is in fact a right triangle because [pic] 5 squared + 12 squared = 13 squared. So the angle labeled [pic] x degrees is a right angle; that is, x = 90. In this case, Quantity A, x, is equal to Quantity B, 90.

Now consider another value of y, say y = 1, to see if the triangle is still a right triangle in this case. If y = 1, then the sides of the triangle have lengths 6, 11, and 13. This triangle is not a right triangle because [pic] 6 squared + 11 squared is not equal to 13 squared. So the angle labeled [pic] x degrees is not a right angle; that is, [pic] x is not equal to 90. In this case, Quantity A, x, is not equal to Quantity B, 90.

Because Quantity A is equal to Quantity B in one case and Quantity A is not equal to Quantity B in another case, the correct answer is Choice D.

This explanation uses the following strategies.

Strategy 10: Trial and Error

Strategy 13: Determine Whether a Conclusion Follows from the Information Given

Question 5.

Refer to the figure.

[pic]

Figure for Geometry Question 5

The figure shows triangle R S T inscribed in a circle, where the vertices R, S, and T lie on the circle clockwise.

In the figure, triangle R S T is inscribed in a circle. The measure of angle R S T is greater than [pic] 90 degrees, and the area of the circle is [pic] 25 pi.

Quantity A: The length of line segment R T

Quantity B: 10

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.

Explanation for Question 5.

Since the area of the circle is [pic] 25 pi, it follows that the radius of the circle is 5 and the diameter is 10. Line segment R T is a diameter of the circle if and only if angle R S T is a right angle. Since you are given that the measure of angle R S T is greater than [pic] 90 degrees, it follows that angle R S T is not a right angle and that line segment R T is a chord but not a diameter. Therefore, Quantity A, the length of line segment R T, is less than Quantity B, 10, and the correct answer is Choice B.

This explanation uses the following strategy.

Strategy 8: Search for a Mathematical Relationship

Question 6.

This question has five answer choices, labeled A through E. Select the best one of the answer choices given.

A construction company will produce identical metal supports in the shape of a right triangle with legs of length 3 feet and 4 feet. The three sides of each triangular support are to be constructed of metal stripping. If the company has a total of 6,000 feet of metal stripping and there is no waste of material in the construction of the supports, what is the greatest possible number of supports that the company can produce?

A. 428

B. 500

C. 545

D. 600

E. 1,000

Select and indicate the best one of the answer choices given.

Explanation for Question 6.

Since each support is in the shape of a right triangle with legs of length 3 feet and 4 feet, the length of the third side of the support is [pic] the square root of 3 squared + 4 squared, end root, or 5 feet. The total length of the stripping of each support is therefore 3 + 4 + 5, or 12 feet. The company has 6,000 feet of metal stripping available. So, with no waste, the greatest possible number of supports that can be produced is [pic] the fraction 6,000 over 12, or 500. The correct answer is Choice B, 500.

This explanation uses the following strategy.

Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

Question 7.

This question has five answer choices, labeled A through E. Select the best one of the answer choices given.

Refer to the figure.

[pic]

Figure for Geometry Question 7

The figure for question 7 shows right triangle A B C, with right angle A between vertical leg A B and horizontal leg A C.

Begin skippable part of figure description.

In the right triangle A B C, vertical leg A B is of length 6 and angle C measures [pic] 30 degrees.

End skippable part of figure description.

What is the area of triangle A B C shown in the figure?

A. 18

B. 20

C. [pic] 12 times the square root of 3

D. [pic] 18 times the square root of 3

E. 36

Select and indicate the best one of the answer choices given.

Explanation for Question 7.

The triangle in the figure accompanying the question is a [pic] 30, 60, 90 degrees triangle, so the ratio of the lengths of the legs is 1 to [pic] the square root of 3. Since the length of the shorter leg, A B, is 6, it follows that the length of the longer leg, A C, is [pic] 6 times the square root of 3. The area of the triangle is therefore [pic] one half times 6, times 6, times the square root of 3, or 18 times the square root of 3. The correct answer is Choice D, [pic] 18 times the square root of 3.

This explanation uses the following strategies.

Strategy 4: Translate from a Figure to an Arithmetic or Algebraic Representation

Strategy 8: Search for a Mathematical Relationship

Question 8.

This question has five answer choices, labeled A through E. Select the best one of the answer choices given.

The volume V of a right circular cylinder is [pic], V = pi, r squared, h, where r is the radius of the base and h is the height of the cylinder. If the volume of a right circular cylinder is [pic] 45 pi and its height is 5, what is the circumference of its base?

A. 3

B. 9

C. [pic] 3 pi

D. [pic] 6 pi

E. [pic] 9 pi

Select and indicate the best one of the answer choices given.

Explanation for Question 8.

It is given that the volume of the right circular cylinder is [pic] 45 pi and the height is 5. It follows that [pic] pi, r squared, h, = 45 pi, or [pic] r squared, h = 45. Since h = 5 and [pic] r squared, h, = 45, it follows that [pic] r squared = 9, or r = 3. Therefore the circumference of the circular base is [pic] 2 pi, r equals 2 pi, times 3, which is equal to 6 pi, and the correct answer is Choice D, [pic] 6 pi.

This explanation uses the following strategy.

Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

Question 9.

This question has five answer choices, labeled A through E. Select the best one of the answer choices given.

Refer to the figure.

[pic]

Figure for Geometry Question 9

The figure for question 9 shows a square inscribed in a circle.

Begin skippable part of figure description.

In the figure, a square is inscribed in a circle; that is, the four vertices of the square lie on the circle. The inscribed square partitions the circle into five regions: the square and four identical,

non-overlapping regions, each of which is inside the circle, but outside the square. One of the four identical, non-overlapping regions is shaded.

End skippable part of figure description.

In the figure, if the square inscribed in the circle has an area of 16, what is the area of the shaded region?

A. [pic] 2 pi, minus one

B. [pic] 2 pi, minus 4

C. [pic] 4 pi, minus 2

D. [pic] 4 pi, minus 4

E. [pic] 8 pi, minus 4

Select and indicate the best one of the answer choices given.

Explanation for Question 9.

It is clear from the figure accompanying the question that the area of the shaded region is [pic] one fourth of the difference between the area of the circle and the area of the square. You are given that the area of the square is 16, so each side has length 4.

You can find the area of the circle if you know the radius of the circle. If you draw a diagonal of the square, as shown in the following figure, you can see that the diagonal is also a diameter of the circle.

[pic]

Figure for Explanation for Geometry Question 9

Begin skippable part of figure description.

In this figure the diagonal of the square has been added to the figure accompanying Question 9, and each side of the square has been labeled 4.

End skippable part figure description.

Note that the diagonal divides the square into two isosceles right triangles with legs of length 4. By the Pythagorean theorem applied to one of the right triangles, the length of the diagonal is equal to [pic] the square root of, 4 squared + 4 squared, end root, or 4 times the square root of 2. Thus the radius of the circle is [pic] r is equal to the fraction with numerator 4 times the square root of 2, and denominator 2, which is equal to 2 times the square root of 2, and the area of the circle is [pic] pi, r squared, = pi, times, open parenthesis, 2, times the square root of 2, close parenthesis, squared, which is equal to 8 pi.

Therefore the area of the shaded region is [pic] the fraction with numerator 8 pi, minus 16, and denominator 4, or 2 pi, minus 4. The correct answer is Choice B, [pic] 2 pi, minus 4.

This explanation uses the following strategies.

Strategy 4: Translate from a Figure to an Arithmetic or Algebraic Representation

Strategy 6: Add to a Geometric Figure

Strategy 8: Search for a Mathematical Relationship

Question 10.

This question has five answer choices, labeled A through E. Select the best one of the answer choices given.

The radius of circle A is r, and the radius of circle B is [pic] 3r over 4. What is the ratio of the area of circle A to the area of circle B ?

A. 1 to 4

B. 3 to 4

C. 4 to 3

D. 9 to 16

E. 16 to 9

Select and indicate the best one of the answer choices given.

Explanation for Question 10.

Circle A has radius r, so its area is [pic] pi, r squared. Circle B has radius [pic] 3 r over 4, so its area is [pic] pi, times, open parenthesis, 3 r over 4, close parenthesis, squared, or the fraction 9 pi, r squared over 16. Therefore the ratio of the area of circle A to the area of circle B is [pic] pi, r squared, to the fraction 9 pi, r squared, over 16, which is the same as the ratio 1 to [pic] 9 over 16, which is the same as the ratio 16 to 9. The correct answer is Choice E, 16 to 9.

This explanation uses the following strategy.

Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

Question 11.

This question has eight answer choices, labeled A through H. Select the best one of the answer choices given.

A flat, rectangular flower bed with an area of 2,400 square feet is bordered by a fence on three sides and by a walkway on the fourth side. If the entire length of the fence is 140 feet, which of the following could be the length, in feet, of one of the sides of the flower bed?

Indicate all such lengths.

A. 20

B. 30

C. 40

D. 50

E. 60

F. 70

G. 80

H. 90

Select and indicate all the answer choices that apply. The correct answer to a question of this type could consist of as few as one or as many as all eight of the answer choices.

Explanation for Question 11.

You know that the area of the rectangular flower bed is 2,400 square feet. So if the flower bed is a feet long and b feet wide, then a b = 2,000. If the side of the flower bed that is bordered by the walkway is one of the sides that are b feet long, then the total length of the three sides of the flower bed bordered by the fence is 2a + b feet. Since it is given that the total length of the fence is 140 feet, it follows that 2a + b = 140. Since a b = 2,000, you can substitute [pic] the fraction 2,400 over a for b in the equation 2a + b = 140 to get the equation [pic] 2a + the fraction 2,400 over a = 140. It follows that [pic] 2, a squared + 2,400 = 140a, or [pic] a squared, minus 70a, + 1,200 = 0.

When you solve this equation for a (either by factoring or by using the quadratic formula), you get a = 30 or a = 40. If a = 30, then [pic] b is equal to the fraction 2,400 over 30, which is equal to 80; if a = 40, then [pic] b is equal to the fraction 2,400 over 40, which is equal to 60. So the possible lengths of the sides are 30, 40, 60, and 80. Thus the correct answer consists of Choices B, C, E, and G; that is, 30, 40, 60, and 80.

This explanation uses the following strategies.

Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

Strategy 8: Search for a Mathematical Relationship

Question 12.

This question has four answer choices, labeled A through D. Select all the answer choices that apply.

Refer to the figure.

[pic]

Figure for Geometry Question 12

Figure for Geometry Question 12

The figure for question 12 shows the four quadrants [pic] 1, 2, 3, and 4 of the x y-plane.

Begin skippable part of figure description.

In the figure, the region above the x-axis and to the right of the y-axis is labeled [pic] 1; the region above the x-axis and to the left of the y-axis is labeled [pic] 2; the region below the x-axis and to the left of the y axis is labeled [pic] 3; and, the region below the x-axis and to the right of the y-axis is labeled [pic] 4.

End skippable part of figure description.

The quadrants of the x y-plane are shown in the figure. In the x y-plane, line m (not shown) has a positive slope and a positive x-intercept. Line m intersects which of the quadrants?

Indicate all such quadrants.

A. Quadrant [pic] 1

B. Quadrant [pic] 2

C. Quadrant [pic] 3

D. Quadrant [pic] 4

Select and indicate all the answer choices that apply. The correct answer to a question of this type could consist of as few as one, or as many as all four of the answer choices.

Explanation for Question 12.

Since line m has a positive x-intercept, it must cross the x-axis to the right of the origin; and since the slope of line m is positive, the line must slant upward and to the right. Consequently, the line must have a negative y-intercept. The figure shows a typical line that satisfies these conditions.

[pic]

Figure for Explanation for Geometry Question 12

Begin skippable part of figure description.

The figure for the explanation of question 12 shows line m in the x y-plane.

In the figure, line m starts in quadrant [pic] 3, and slants upward and to the right. It first crosses the y-axis below the origin, passing through quadrant [pic] 4, and then crosses the x-axis to the right of the origin. Line m extends into quadrant [pic] 1.

End skippable part of figure description.

In the figure, the line intersects quadrants [pic] 1, 3, and 4. Thus the correct answer consists of Choices A, C, and D; that is, quadrant [pic] 1, quadrant [pic] 3, and quadrant [pic] 4.

This explanation uses the following strategy.

Strategy 6: Add to a Geometric Figure

Question 13.

This question does not have any answer choices; it is a numeric entry question. To answer this question, enter a number in the answer space provided.

Refer to the figure.

[pic]

Figure for Geometry Question 13

The figure for question 13 shows a horizontal line and a slanted line, beginning at a point on the horizontal line, and extending upward and to the right forming two adjacent angles at the point where the two lines intersect.

Begin skippable part of figure description.

The angle above the horizontal line and to the left of the slanted line measures r degrees, and the angle above the horizontal line and to the right of the slanted line measures s degrees.

End skippable part of figure description.

In the figure, if [pic] the fraction r over r + s is equal to the fraction 5 over 8, what is the value of r ?

The answer space is preceded by the label “r =.”

To answer this question, enter a number in the answer space provided. The number can include a decimal point, and can be positive, negative, or zero. The number entered cannot be a fraction.

Explanation for Question 13.

From the figure, note that [pic] r degrees + s degrees must equal [pic] 180 degrees. Therefore [pic] the fraction r over r + s is equal to the fraction r over 180. Since you are also given in the question that [pic] the fraction r over r + s is equal to the fraction 5 over 8, you can conclude that [pic] the fraction r over 180 is equal to the fraction 5 over 8. Thus [pic] r is equal to the fraction with numerator 5 times 180 and denominator 8, which is equal to 112.5, and the correct answer is 112.5.

This explanation uses following strategy.

Strategy 4: Translate from a Figure to an Arithmetic or Algebraic Representation

This is the end of Chapter 5 – Geometry Answer Key with Answers and Explanations.

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