GRADUATE RECORD EXAMINATIONS®
Official GRE 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. 18 times the square root of 3
Question 8.
Answer: Choice D. 6 pi
Question 9.
Answer: Choice B. 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 1
Choice C. Quadrant 3
Choice D. Quadrant 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 with coordinates 2 comma 2. The diagonals of S intersect at the point 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 with coordinates 2 comma 2 is a vertex of square S and the point with coordinates 6 comma 6 is the midpoint of the diagonals, it follows that the point 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 with coordinates 2 comma 2, the point with coordinates 6 comma 6, and the point with coordinates 10 comma 10, as shown in the following figure.
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 with coordinates 2 comma 2, is the lower-left vertex of the square and the point with coordinates 10 comma 10 is the upper-right vertex. The lower-right vertex is the point with coordinates 10 comma 2, and the upper-left vertex is the point with coordinates 2 comma 10.
The intersection of the diagonals of the square is at the point 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 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 one fifth of its perimeter. Thus Quantity A, the length of a side of a regular pentagon with a perimeter of 12.5, is 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 one sixth of its perimeter. Thus Quantity B, the length of a side of a regular hexagon with a perimeter of 15, is 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 with coordinates 5 comma 4 and the point 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 with coordinates 5 comma 4 and the point with coordinates 2 comma negative one on the line, as shown in the figure.
Figure for Explanation for Geometry Question 3
Begin skippable part of figure description.
In the figure, the point with coordinates 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 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.
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 12 minus y, and hypotenuse of length 13. The angle that appears to be a right angle is labeled 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 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 5 squared + 12 squared = 13 squared. So the angle labeled 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 6 squared + 11 squared is not equal to 13 squared. So the angle labeled x degrees is not a right angle; that is, 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.
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 90 degrees, and the area of the circle is 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 25 pi, it follows that the radius of the circle is 5 and the diameter is 10. Linesegment 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 90 degrees, it follows that angle R S T is not a right angle and that linesegment 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 theshape of a right triangle with legsof length 3 feet and 4 feet. The threesides 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 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 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.
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 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. 12 times the square root of 3
D. 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 30, 60, 90 degrees triangle, so the ratio of the lengths of the legs is 1 to 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 6 times the square root of 3. The area of the triangle is therefore one half times 6, times 6, times the square root of 3, or 18 times the square root of 3. The correct answer is ChoiceD, 18 times the square root of 3.