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More Lessons for the Regents High School Exam

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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 August 2019.

The following are questions from the past paper Regents High School Geometry, August 2019 Exam (pdf). Scroll down the page for the step by step solutions.

Geometry - August 2019 Regents - Questions and solutions 1 - 12

- 1 On the set of axes below, AB is dilated by a scale factor of 5/2 centered at point P.
- The coordinates of the vertices of parallelogram CDEH are C(-5,5), D(2,5), E(-1,-1), and H(-8,-1). What are the coordinates of P, the point of intersection of diagonals CE and DH?
- The coordinates of the endpoints of QS are Q(-9,8) and S(9,-4). Point R is on QS such that QR:RS is in the ratio of 1:2. What are the coordinates of point R?
- If the altitudes of a triangle meet at one of the triangle’s vertices, then the triangle is
- In the diagram below of triangle ACD, DB is a median to AC, and AB ≅ DB
- What are the coordinates of the center and the length of the radius of the circle whose equation is x
^{2}+ y^{2}= 8x - 6y + 39?

- In the diagram below of parallelogram ABCD, AFGB, CF bisects ∠DCB, DG bisects ∠ADC, and CF and DG intersect at E.
- What is an equation of a line that is perpendicular to the line whose equation is 2y + 3x = 1?
- Triangles ABC and RST are graphed on the set of axes below
- If the line represented by y = -1/4 x - 2 is dilated by a scale factor of 4 centered at the origin, which statement about the image is true?
- Square MATH has a side length of 7 inches. Which three-dimensional object will be formed by continuously rotating square MATH around side AT? 12. Circle O with a radius of 9 is drawn below. The measure of central angle AOC is 120°.

Geometry - August 2019 Regents - Questions and solutions 13 - 24

13. In quadrilateral QRST, diagonals QS and RT intersect at M. Which statement would always prove quadrilateral QRST is a parallelogram?

14. A standard-size golf ball has a diameter of 1.680 inches. The material used to make the golf ball weighs 0.6523 ounce per cubic inch. What is the weight, to the nearest hundredth of an ounce, of one golf ball?

15. Chelsea is sitting 8 feet from the foot of a tree. From where she is sitting, the angle of elevation of her line of sight to the top of the tree is 36°. If her line of sight starts 1.5 feet above ground, how tall is the tree, to the nearest foot?

16. 6 In the diagram below of right triangle ABC, altitude CD intersects hypotenuse AB at D.

17. A countertop for a kitchen is modeled with the dimensions shown below. An 18-inch by 21-inch rectangle will be removed for the installation of the sink.

18. In the diagram below, BC connects points B and C on the congruent sides of isosceles triangle ADE, such that rABC is isosceles with vertex angle A.

- In triangle ABC below, angle C is a right angle.
- In right triangle RST, altitude TV is drawn to hypotenuse RS. If RV = 12 and RT = 18, what is the length of SV?
- What is the volume, in cubic centimeters, of a right square pyramid with base edges that are 64 cm long and a slant height of 40 cm?
- In the diagram below, chords PQ and RS of circle O intersect at T.
- A rhombus is graphed on the set of axes below. Which transformation would carry the rhombus onto itself?
- A 15-foot ladder leans against a wall and makes an angle of 65° with the ground. What is the horizontal distance from the wall to the base of the ladder, to the nearest tenth of a foot?

Geometry - August 2019 Regents - Questions and solutions Parts 2-4

25. In parallelogram ABCD shown below, m∠DAC = 98° and m∠ACD = 36°. What is the measure of angle B? Explain why.

26. An airplane took off at a constant angle of elevation. After the plane traveled for 25 miles, it reached an altitude of 5 miles, as modeled below.

To the nearest tenth of a degree, what was the angle of elevation?

27. On the set of axes below, triangle ABC ≅ triangle DEF. Describe a sequence of rigid motions that maps triangle ABC onto triangle DEF.

- The vertices of triangle ABC have coordinates A(-2,-1), B(10,-1), and C(4,4). Determine and state the area of triangle ABC. [The use of the set of axes below is optional.]
- Using the construction below, state the degree measure of ∠CAD. Explain why.
- In the diagram below of circle K, secant PLKE and tangent PZ are drawn from external point P. If mLZ° = 56°, determine and state the degree measure of angle P.
- A large water basin is in the shape of a right cylinder. The inside of the basin has a diameter of 8 1/4 feet and a height of 3 feet. Determine and state, to the nearest cubic foot, the number of cubic feet of water that it will take to fill the basin to a level of 1/2 foot from the top.

- Triangle ABC is shown below. Using a compass and straightedge, construct the dilation of rABC centered at B with a scale factor of 2. [Leave all construction marks.]
- In the diagram below, triangle ABE ≅ triangle CBD.
- A cargo trailer, pictured below, can be modeled by a rectangular prism and a triangular prism. Inside the trailer, the rectangular prism measures 6 feet wide and 10 feet long. The walls that form the triangular prism each measure 4 feet wide inside the trailer. The diagram below is of the floor, showing the inside measurements of the trailer.

If the inside height of the trailer is 6.5 feet, what is the total volume of the inside of the trailer, to the nearest cubic foot?

- The coordinates of the vertices of rABC are A(1,2), B(-5,3), and C(-6,-3).

Prove that triangle ABC is isosceles.

[The use of the set of axes on the next page is optional.]

State the coordinates of point D such that quadrilateral ABCD is a square.

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