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More Lessons for the Regents High School Exam

More Lessons for Algebra

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 2018.

### Geometry Common Core Regents New York State Exam - January 2018

The following are questions from the past paper
Regents High School Geometry, January 2018 Exam (pdf). Scroll down the page for the step by step solutions.

Geometry - January 2018 Regents - Questions and solutions 1 - 12

1. In the diagram below, a sequence of rigid motions maps ABCD onto JKLM.

If m∠A 82°, m∠B 104°, and m∠L 121°, the measure of ∠M is

2. Parallelogram HAND is drawn below with diagonals HN and AD intersecting at S. Which statement is always true?

3. The graph below shows two congruent triangles, ABC and A'B'C'. Which rigid motion would map ABC onto A'B'C'?

4. A man was parasailing above a lake at an angle of elevation of 32° from a boat, as modeled in the diagram below.

If 129.5 meters of cable connected the boat to the parasail, approximately how many meters above the lake was the man?

5. A right hexagonal prism is shown below. A two-dimensional cross section that is perpendicular to the base is taken from the prism.

Which figure describes the two-dimensional cross section?

6. In the diagram below, has endpoints with coordinates A(-5,2) and C(4,-10).

If B is a point on and AB:BC = 1:2, what are the coordinates of B?

7. An ice cream waffle cone can be modeled by a right circular cone with a base diameter of 6.6 centimeters and a volume of 54.45π cubic centimeters. What is the number of centimeters in the height of the waffle cone?

8. The vertices of PQR have coordinates P(2,3), Q(3,8), and R(7,3).

Under which transformation of PQR are distance and angle measure preserved?

9. In ABC shown below, side AC is extended to point D with m∠DAB = (180 - 3x)°, m∠B = (6x - 40)°, and m∠C = (x + 20)°. What is m∠BAC?

10. Circle O is centered at the origin. In the diagram below, a quarter of circle O is graphed.

Which three-dimensional figure is generated when the quarter circle is continuously rotated about the y-axis?

11. Rectangle A'B'C'D' is the image of rectangle ABCD after a dilation centered at point A by a scale factor of 2/3. Which statement is correct?

12. The equation of a circle is x^{2} + y^{2} - 6x + 2y = 6. What are the
coordinates of the center and the length of the radius of the circle?

Geometry - January 2018 Regents - Questions and solutions 13 - 24

13. 3 In the diagram of ABC below, DE is parallel to AB, CD = 15, AD = 9, and AB = 40. The length of DE is

14. The line whose equation is 3x - 5y = 4 is dilated by a scale factor 5/3 of centered at the origin. Which statement is correct?

15. Which transformation would not carry a square onto itself?

16. In circle M below, diameter AC, chords AB and BC, and radius MB are drawn. Which statement is not true?

17. In the diagram below, XS and YR intersect at Z. Segments XY and RS are drawn perpendicular to YR to form triangles XYZ and SRZ. Which statement is always true?

18. As shown in the diagram below, ABC || EFG and BF ≅ EF. If m∠CBF = 42.5deg;, then m∠BF is

19. A parallelogram must be a rhombus if its diagonals

20. What is an equation of a line which passes through (6,9) and is perpendicular to the line whose equation is 4x - 6y = 15?

21. Quadrilateral ABCD is inscribed in circle O, as shown below.

If m∠A = 80°, m∠B = 75°, m∠C = (y + 30)°, and m∠D = (x - 10)°,

which statement is true?

22. A regular pyramid has a square base. The perimeter of the base is 36 inches and the height of the pyramid is 15 inches. What is the volume of the pyramid in cubic inches?

23. In the diagram below of ABC, ∠ABC is a right angle, AC = 12, AD = 8, and altitude BD is drawn

What is the length of BC?

24. In the diagram below, two concentric circles with center O, and radii OC, OD, OCE, and ODF are drawn. If OC = 4 and OE = 6, which relationship between the length of arc EF and the length of arc CD is always true?

Geometry - January 2018 Regents - Questions and solutions Parts 2-4

25. Given: Parallelogram ABCD with diagonal AC drawn Prove: ABC ≅ CDA

26. The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

27. Given: Right triangle ABC with right angle at C

If sin A increases, does cos B increase or decrease? Explain why

28. In the diagram below, the circle has a radius of 25 inches. The area of the unshaded sector is 500π in^{2}.
Determine and state the degree measure of angle Q, the central angle of the shaded sector.

29. A machinist creates a solid steel part for a wind turbine engine. The part has a volume of 1015 cubic centimeters. Steel can be purchased for $0.29 per kilogram, and has a density of 7.95 g/cm^{3}.
If the machinist makes 500 of these parts, what is the cost of the steel, to the nearest dollar?

30. In the graph below, ABC has coordinates A(-9,2), B(-6,-6), and C(-3,-2), and RST has coordinates R(-2,9), S(5,6), and T(2,3).

Is ABC congruent to RST? Use the properties of rigid motions to explain your reasoning.

31. Bob places an 18-foot ladder 6 feet from the base of his house and leans it up against the side of his house. Find, to the nearest degree, the measure of the angle the bottom of the ladder makes with the ground.

32. Triangle ABC and triangle ADE are graphed on the set of axes below.

Describe a transformation that maps triangle ABC onto triangle ADE.

Explain why this transformation makes triangle ADE similar to triangle ABC.

33. A storage tank is in the shape of a cylinder with a hemisphere on the top. The highest point on the inside of the storage tank is 13 meters above the floor of the storage tank, and the diameter inside the cylinder is 8 meters. Determine and state, to the nearest cubic meter, the total volume inside the storage tank.

34. As shown in the diagram below, an island (I) is due north of a marina (M). A boat house (H) is 4.5 miles due west of the marina. From the boat house, the island is located at an angle of 54° from the marina.

Determine and state, to the nearest tenth of a mile, the distance from the boat house (H) to the island (I).

Determine and state, to the nearest tenth of a mile, the distance from the island (I) to the marina (M).

35. In the coordinate plane, the vertices of triangle PAT are P(-1,-6), A(-4,5), and T(5,-2). Prove that PAT is an isosceles triangle. [The use of the set of axes on the next page is optional.]

State the coordinates of R so that quadrilateral PART is a parallelogram.

Prove that quadrilateral PART is a parallelogram.

More Lessons for the Regents High School Exam

More Lessons for Algebra

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 2018.

Geometry - January 2018 Regents - Questions and solutions 1 - 12

1. In the diagram below, a sequence of rigid motions maps ABCD onto JKLM.

If m∠A 82°, m∠B 104°, and m∠L 121°, the measure of ∠M is

2. Parallelogram HAND is drawn below with diagonals HN and AD intersecting at S. Which statement is always true?

3. The graph below shows two congruent triangles, ABC and A'B'C'. Which rigid motion would map ABC onto A'B'C'?

4. A man was parasailing above a lake at an angle of elevation of 32° from a boat, as modeled in the diagram below.

If 129.5 meters of cable connected the boat to the parasail, approximately how many meters above the lake was the man?

5. A right hexagonal prism is shown below. A two-dimensional cross section that is perpendicular to the base is taken from the prism.

Which figure describes the two-dimensional cross section?

6. In the diagram below, has endpoints with coordinates A(-5,2) and C(4,-10).

If B is a point on and AB:BC = 1:2, what are the coordinates of B?

7. An ice cream waffle cone can be modeled by a right circular cone with a base diameter of 6.6 centimeters and a volume of 54.45π cubic centimeters. What is the number of centimeters in the height of the waffle cone?

8. The vertices of PQR have coordinates P(2,3), Q(3,8), and R(7,3).

Under which transformation of PQR are distance and angle measure preserved?

9. In ABC shown below, side AC is extended to point D with m∠DAB = (180 - 3x)°, m∠B = (6x - 40)°, and m∠C = (x + 20)°. What is m∠BAC?

10. Circle O is centered at the origin. In the diagram below, a quarter of circle O is graphed.

Which three-dimensional figure is generated when the quarter circle is continuously rotated about the y-axis?

11. Rectangle A'B'C'D' is the image of rectangle ABCD after a dilation centered at point A by a scale factor of 2/3. Which statement is correct?

12. The equation of a circle is x

13. 3 In the diagram of ABC below, DE is parallel to AB, CD = 15, AD = 9, and AB = 40. The length of DE is

14. The line whose equation is 3x - 5y = 4 is dilated by a scale factor 5/3 of centered at the origin. Which statement is correct?

15. Which transformation would not carry a square onto itself?

16. In circle M below, diameter AC, chords AB and BC, and radius MB are drawn. Which statement is not true?

17. In the diagram below, XS and YR intersect at Z. Segments XY and RS are drawn perpendicular to YR to form triangles XYZ and SRZ. Which statement is always true?

18. As shown in the diagram below, ABC || EFG and BF ≅ EF. If m∠CBF = 42.5deg;, then m∠BF is

19. A parallelogram must be a rhombus if its diagonals

20. What is an equation of a line which passes through (6,9) and is perpendicular to the line whose equation is 4x - 6y = 15?

21. Quadrilateral ABCD is inscribed in circle O, as shown below.

If m∠A = 80°, m∠B = 75°, m∠C = (y + 30)°, and m∠D = (x - 10)°,

which statement is true?

22. A regular pyramid has a square base. The perimeter of the base is 36 inches and the height of the pyramid is 15 inches. What is the volume of the pyramid in cubic inches?

23. In the diagram below of ABC, ∠ABC is a right angle, AC = 12, AD = 8, and altitude BD is drawn

What is the length of BC?

24. In the diagram below, two concentric circles with center O, and radii OC, OD, OCE, and ODF are drawn. If OC = 4 and OE = 6, which relationship between the length of arc EF and the length of arc CD is always true?

Geometry - January 2018 Regents - Questions and solutions Parts 2-4

25. Given: Parallelogram ABCD with diagonal AC drawn Prove: ABC ≅ CDA

26. The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

27. Given: Right triangle ABC with right angle at C

If sin A increases, does cos B increase or decrease? Explain why

28. In the diagram below, the circle has a radius of 25 inches. The area of the unshaded sector is 500π in

29. A machinist creates a solid steel part for a wind turbine engine. The part has a volume of 1015 cubic centimeters. Steel can be purchased for $0.29 per kilogram, and has a density of 7.95 g/cm

30. In the graph below, ABC has coordinates A(-9,2), B(-6,-6), and C(-3,-2), and RST has coordinates R(-2,9), S(5,6), and T(2,3).

Is ABC congruent to RST? Use the properties of rigid motions to explain your reasoning.

31. Bob places an 18-foot ladder 6 feet from the base of his house and leans it up against the side of his house. Find, to the nearest degree, the measure of the angle the bottom of the ladder makes with the ground.

32. Triangle ABC and triangle ADE are graphed on the set of axes below.

Describe a transformation that maps triangle ABC onto triangle ADE.

Explain why this transformation makes triangle ADE similar to triangle ABC.

33. A storage tank is in the shape of a cylinder with a hemisphere on the top. The highest point on the inside of the storage tank is 13 meters above the floor of the storage tank, and the diameter inside the cylinder is 8 meters. Determine and state, to the nearest cubic meter, the total volume inside the storage tank.

34. As shown in the diagram below, an island (I) is due north of a marina (M). A boat house (H) is 4.5 miles due west of the marina. From the boat house, the island is located at an angle of 54° from the marina.

Determine and state, to the nearest tenth of a mile, the distance from the boat house (H) to the island (I).

Determine and state, to the nearest tenth of a mile, the distance from the island (I) to the marina (M).

35. In the coordinate plane, the vertices of triangle PAT are P(-1,-6), A(-4,5), and T(5,-2). Prove that PAT is an isosceles triangle. [The use of the set of axes on the next page is optional.]

State the coordinates of R so that quadrilateral PART is a parallelogram.

Prove that quadrilateral PART is a parallelogram.

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