High School Math based on the topics required for the Regents
Exam conducted by NYSED. The following are the worked solutions
for the Geometry (Common Core) Regents High School Examination
January 2016.

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### Geometry Common Core Regents New York State Exam - January 2016

The following are questions from the past paper Regents High School Geometry, January 2016 Exam (pdf).

Scroll down the page for the step by step solutions.

Geometry - January 2016 Regents - Questions and solutions 1 - 5

1. William is drawing pictures of cross sections of the right circular cone computations. below.

Which drawing can not be a cross section of a cone?

2. An equation of a line perpendicular to the line represented by the equation y = -\(\frac{1}{2}\)x - 5 and passing through (6,4) is

3. In parallelogram QRST shown below, diagonal TR is drawn, U and V are points on TS and QR, respectively, and UV intersects TR at W

If m∠S 60°, m∠SRT 83°, and m∠TWU 35°, what is m∠WVQ?

4. A fish tank in the shape of a rectangular prism has dimensions of 14 inches, 16 inches, and 10 inches. The tank contains 1680 cubic inches of water. What percent of the fish tank is empty?

5. Which transformation would result in the perimeter of a triangle being different from the perimeter of its image? Geometry - January 2016 Regents - Questions and solutions 6 - 10

6. In the diagram below, FE bisects AC at B, and GE bisects BD at C. Which statement is always true?

7. As shown in the diagram below, a regular pyramid has a square base whose side measures 6 inches.

If the altitude of the pyramid measures 12 inches, its volume, in cubic inches, is

8. Triangle ABC and triangle DEF are graphed on the set of axes below.

Which sequence of transformations maps triangle ABC onto triangle DEF?

(1) a reflection over the x-axis followed by a reflection over the y-axis

(2) a 180° rotation about the origin followed by a reflection over the line y = x

(3) a 90° clockwise rotation about the origin followed by a reflection over the y-axis

(4) a translation 8 units to the right and 1 unit up followed by a 90° counterclockwise rotation about the origin

9. In ABC, the complement of ∠B is ∠A. Which statement is always true?

10. A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line

(1) is perpendicular to the original line

(2) is parallel to the original line

(3) passes through the origin

(4) is the original line

Geometry - January 2016 Regents - Questions and solutions 11 - 15

11. Quadrilateral ABCD is graphed on the set of axes below.

When ABCD is rotated 90° in a counterclockwise direction about the origin, its image is quadrilateral A'B'C'D'. Is distance preserved under this rotation, and which coordinates are correct for the given vertex?

(1) no and C'(1,2) (3) yes and A'(6,2)

(2) no and D'(2,4) (4) yes and B'(3,4)

12. In the diagram below of circle O, the area of the shaded sector LOM is 2π cm^{2}.

If the length of NL is 6 cm, what is m∠N?

13. In the diagram below, △ABC ∼ △DEF.

If AB 6 and AC 8, which statement will justify similarity by SAS?

14. The diameter of a basketball is approximately 9.5 inches and the diameter of a tennis ball is approximately 2.5 inches. The volume of the basketball is about how many times greater than the volume of the tennis ball?

(1) 3591 (3) 55

(2) 65 (4) 4

15. The endpoints of one side of a regular pentagon are (1,4) and (2,3).

What is the perimeter of the pentagon? Geometry - January 2016 Regents - Questions and solutions 16 - 20

16. In the diagram of right triangle ABC shown below, AB = 14 and AC = 9.

What is the measure of ∠A, to the nearest degree?

17. What are the coordinates of the center and length of the radius of the circle whose equation is x^{2} + 6x + y^{2} - 4y = 23?

18. The coordinates of the vertices of △RST are R(2,3), S(8,2), and T(4,5). Which type of triangle is △RST?

(1) right (3) obtuse

(2) acute (4) equiangular

19. Molly wishes to make a lawn ornament in the form of a solid sphere. The clay being used to make the sphere weighs .075 pound per cubic inch. If the sphere’s radius is 4 inches, what is the weight of the sphere, to the nearest pound?

(1) 34 (3) 15

(2) 20 (4) 4

20. The ratio of similarity of △BOY to △GRL is 1:2. If BO = x + 3 and GR = 3x - 1, then the length of GR is

(1) 5 (3) 10

(2) 7 (4) 20

Geometry - January 2016 Regents - Questions and solutions 21 - 24

21. In the diagram below, DC, AC, DOB, CB, and AB are chords of circle O, FDE is tangent at point D, and radius AO is drawn. Sam decides to apply this theorem to the diagram: “An angle inscribed in a semi-circle is a right angle.”

Which angle is Sam referring to?

22. In the diagram below, CD is the altitude drawn to the hypotenuse AB of right triangle ABC.

Which lengths would not produce an altitude that measures \(6\sqrt 2 \)?

23. A designer needs to create perfectly circular necklaces. The necklaces each need to have a radius of 10 cm. What is the largest number of necklaces that can be made from 1000 cm of wire?

24. In SCU shown below, points T and O are on SU and CU, respectively. Segment OT is drawn so that ∠C ≅ ∠OTU. If TU = 4, OU = 5, and OC = 7, what is the length of ST?

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More Lessons for the Regents High School Exam

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Geometry - January 2016 Regents - Questions and solutions 1 - 5

1. William is drawing pictures of cross sections of the right circular cone computations. below.

Which drawing can not be a cross section of a cone?

2. An equation of a line perpendicular to the line represented by the equation y = -\(\frac{1}{2}\)x - 5 and passing through (6,4) is

3. In parallelogram QRST shown below, diagonal TR is drawn, U and V are points on TS and QR, respectively, and UV intersects TR at W

If m∠S 60°, m∠SRT 83°, and m∠TWU 35°, what is m∠WVQ?

4. A fish tank in the shape of a rectangular prism has dimensions of 14 inches, 16 inches, and 10 inches. The tank contains 1680 cubic inches of water. What percent of the fish tank is empty?

5. Which transformation would result in the perimeter of a triangle being different from the perimeter of its image? Geometry - January 2016 Regents - Questions and solutions 6 - 10

6. In the diagram below, FE bisects AC at B, and GE bisects BD at C. Which statement is always true?

7. As shown in the diagram below, a regular pyramid has a square base whose side measures 6 inches.

If the altitude of the pyramid measures 12 inches, its volume, in cubic inches, is

8. Triangle ABC and triangle DEF are graphed on the set of axes below.

Which sequence of transformations maps triangle ABC onto triangle DEF?

(1) a reflection over the x-axis followed by a reflection over the y-axis

(2) a 180° rotation about the origin followed by a reflection over the line y = x

(3) a 90° clockwise rotation about the origin followed by a reflection over the y-axis

(4) a translation 8 units to the right and 1 unit up followed by a 90° counterclockwise rotation about the origin

9. In ABC, the complement of ∠B is ∠A. Which statement is always true?

10. A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line

(1) is perpendicular to the original line

(2) is parallel to the original line

(3) passes through the origin

(4) is the original line

11. Quadrilateral ABCD is graphed on the set of axes below.

When ABCD is rotated 90° in a counterclockwise direction about the origin, its image is quadrilateral A'B'C'D'. Is distance preserved under this rotation, and which coordinates are correct for the given vertex?

(1) no and C'(1,2) (3) yes and A'(6,2)

(2) no and D'(2,4) (4) yes and B'(3,4)

12. In the diagram below of circle O, the area of the shaded sector LOM is 2π cm

If the length of NL is 6 cm, what is m∠N?

13. In the diagram below, △ABC ∼ △DEF.

If AB 6 and AC 8, which statement will justify similarity by SAS?

14. The diameter of a basketball is approximately 9.5 inches and the diameter of a tennis ball is approximately 2.5 inches. The volume of the basketball is about how many times greater than the volume of the tennis ball?

(1) 3591 (3) 55

(2) 65 (4) 4

15. The endpoints of one side of a regular pentagon are (1,4) and (2,3).

What is the perimeter of the pentagon? Geometry - January 2016 Regents - Questions and solutions 16 - 20

16. In the diagram of right triangle ABC shown below, AB = 14 and AC = 9.

What is the measure of ∠A, to the nearest degree?

17. What are the coordinates of the center and length of the radius of the circle whose equation is x

18. The coordinates of the vertices of △RST are R(2,3), S(8,2), and T(4,5). Which type of triangle is △RST?

(1) right (3) obtuse

(2) acute (4) equiangular

19. Molly wishes to make a lawn ornament in the form of a solid sphere. The clay being used to make the sphere weighs .075 pound per cubic inch. If the sphere’s radius is 4 inches, what is the weight of the sphere, to the nearest pound?

(1) 34 (3) 15

(2) 20 (4) 4

20. The ratio of similarity of △BOY to △GRL is 1:2. If BO = x + 3 and GR = 3x - 1, then the length of GR is

(1) 5 (3) 10

(2) 7 (4) 20

21. In the diagram below, DC, AC, DOB, CB, and AB are chords of circle O, FDE is tangent at point D, and radius AO is drawn. Sam decides to apply this theorem to the diagram: “An angle inscribed in a semi-circle is a right angle.”

Which angle is Sam referring to?

22. In the diagram below, CD is the altitude drawn to the hypotenuse AB of right triangle ABC.

Which lengths would not produce an altitude that measures \(6\sqrt 2 \)?

23. A designer needs to create perfectly circular necklaces. The necklaces each need to have a radius of 10 cm. What is the largest number of necklaces that can be made from 1000 cm of wire?

24. In SCU shown below, points T and O are on SU and CU, respectively. Segment OT is drawn so that ∠C ≅ ∠OTU. If TU = 4, OU = 5, and OC = 7, what is the length of ST?

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