Gre chapter 5 geometry answers explanations 2 PDF

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GRADUATE RECORD EXAMINATIONS®

Official GRE Quantitative Reasoning Practice Questions, Volume 1 Chapter 5 – Geometry Answer Key with Answers and Explanations

Copyright© 2014 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS, and G RE are registered trademarks of Educational Testing Service (ETS) in the United States and other countries. c3c02e1c5c3b58c7423ffdf264998e96.doc

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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

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Question 8. Answer: Choice D.

6 pi

Question 9. 2 pi, minus 4

Answer: Choice B.

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

Choice C.

Quadrant

3

Choice D.

Quadrant

4

1

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

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Answers and Explanations Question 1. In the xy-plane, one of the vertices of square S is the point The diagonals of S intersect at the point

with coordinates 2 comma 2.

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 point

with coordinates 2 comma 2, the

with coordinates 6 comma 6, and the point

10, as shown in the following figure.

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with coordinates 10 comma

F i g u r ef o rEx p l a na t i o nf o rGe o me t r yQu e s t i o n1

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 and the point vertex is the point

with coordinates 2 comma 2, is the lower-left vertex of the square

with coordinates 10 comma 10 is the upper-right vertex. The lower-right 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

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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

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Question 3. It is given that a line in the xy-plane contains the point point

with coordinates 5 comma 4 and the

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 xy-plane and labeling the point 5 comma 4 and the point

with coordinates 2 comma negative one on the line, as shown in

the figure.

F i g u r ef o rEx p l a na t i o nf o rGe o me t r yQu e s t i o n3 c3c02e1c5c3b58c7423ffdf264998e96.doc

with coordinates

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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.

F i g u r ef o rGe o me t r yQu e s t i on4 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.

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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

x degrees is a right angle; that is, x = 90. In this squared = 13 squared. So the angle labeled 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, not equal to Quantity B, 90.

x is not equal to 90. In this case, Quantity A, x, is

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

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Question 5. Refer to the figure.

F i g u r ef o rGe o me t r yQu e s t i on5 The figure shows triangle RST inscribed in a circle, where the vertices R, S, and T lie on the circle clockwise. In the figure, triangle RST is inscribed in a circle. The measure of angle RST is greater than 90 degrees, and the area of the circle is

25 pi.

Quantity A: The length of line segment RT 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.

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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. Line segment RT is a diameter of the circle if and only if angle RST is a right angle. Since you are given that the measure of angle RST is greater than 90 degrees, it follows that angle RST is not a right angle and that line segment RT is a chord but not a diameter. Therefore, Quantity A, the length of line segment RT, 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.

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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 the fraction 6,000 over 12, or

possible number of supports that can be produced is 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.

F i g u r ef o rGe o me t r yQu e s t i on7 The figure for question 7 shows right triangle ABC, with right angle A between vertical leg AB and horizontal leg AC.

Begin skippable part of figure description. In the right triangle ABC, vertical leg AB is of length 6 and angle C measures

End skippable part of figure description.

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30 degrees.

What is the area of triangle ABC 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 degrees triangle, so the ratio of the lengths of the legs is 1 to

30, 60, 90 the square root of 3. Since the

length of the shorter leg, AB, is 6, it follows that the length of the longer leg, AC, is times the square root of 3. The area of the triangle is therefore

6 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,

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

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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

, 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 and its height is 5, what is the circumference of its base? A.

3

B.

9

C.

3 pi

D.

6 pi

E.

9 pi

45 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 that

pi, r squared, h, = 45 pi, or

r squared, h = 45. Since h = 5 and

r squared, h, = 45, it follows that

r squared = 9, or r = 3. Therefore the

circumference of the circular base is to 6 pi, and the correct answer is Choice D,

45 pi and the height is 5. It follows

2 pi, r equals 2 pi, times 3, which is equal 6 pi.

This explanation uses the following strategy. Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

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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.

F i g u r ef o rGe o me t r yQu e s t i on9 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.

2 pi, minus one

B.

2 pi, minus 4

C.

4 pi, minus 2

D.

4 pi, minus 4

E.

8 pi, minus 4

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

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Explanation for Question 9. It is clear from the figure accompanying the question that the area of the shaded region is 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.

F i g u r ef o rEx p l a na t i o nf o rGe o me t r yQu e s t i o n9 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 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

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

pi, r squared, = pi, times, open parenthesis, 2, times

the square root of 2, close parenthesis, squared, which is equal to 8 pi.

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Therefore the area of the shaded region is

the fraction with numerator 8 pi,

minus 16, and denominator 4, or 2 pi, minus 4. The correct answer is Choice B, minus 4.

2 pi,

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. 3r over 4. What is the ratio of the area

The radius of circle A is r, and the radius of circle B is 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 give...