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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 June 2019.

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

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

- On the set of axes below, triangle ABC is graphed. Triangles A’B’C’ and A"B"C", the images of triangle ABC, are graphed after a sequence of rigid motions.
- The table below shows the population and land area, in square miles, of four counties in New York State at the turn of the century.
- If a rectangle is continuously rotated around one of its sides, what is the three-dimensional figure formed?
- Which transformation carries the parallelogram below onto itself?
- After a dilation centered at the origin, the image of CD is CD. If the coordinates of the endpoints of these segments are C(6,-4), D(2,-8), C(9,-6), and D(3,-12), the scale factor of the dilation is
- A tent is in the shape of a right pyramid with a square floor. The square floor has side lengths of 8 feet. If the height of the tent at its center is 6 feet, what is the volume of the tent, in cubic feet?

- The line -3x + 4y = 8 is transformed by a dilation centered at the origin. Which linear equation could represent its image?
- In the diagram below, AC and BD intersect at E.
- The expression sin 57° is equal to
- What is the volume of a hemisphere that has a diameter of 12.6 cm, to the nearest tenth of a cubic centimeter?
- In the diagram below of triangle ABC, D is a point on BA, E is a point on BC, and DE is drawn
- A quadrilateral must be a parallelogram if

(1) one pair of sides is parallel and one pair of angles is congruent

(2) one pair of sides is congruent and one pair of angles is congruent

(3) one pair of sides is both parallel and congruent

(4) the diagonals are congruent

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

13. In the diagram below of circle O, chords JT and ER intersect at M.

14. Triangles JOE and SAM are drawn such that ∠E ≅ ∠M and EJ ≅ MS. Which mapping would not always lead to triangle JOE ≅ triangle SAM?

15. In triangle ABC shown below, ∠ACB is a right angle, E is a point on AC, and ED is drawn perpendicular to hypotenuse AB.

16. Which equation represents a line parallel to the line whose equation is -2x + 3y = -4 and passes through the point (1,3)?

17. In rhombus TIGE, diagonals TG and IE intersect at R. The perimeter of TIGE is 68, and TG = 16. What is the length of diagonal IE?

18. In circle O two secants, ABP and CDP, are drawn to external point P. If mAC = 72°, and mBD = 34°, what is the measure of ∠P?

- What are the coordinates of point C on the directed segment from A(-8,4) to B(10,-2) that partitions the segment such that AC:CB is 2:1?
- The equation of a circle is x
^{2}+ 8x + y^{2}- 12y = 144. What are the coordinates of the center and the length of the radius of the circle? - In parallelogram PQRS, QP is extended to point T and ST is drawn.
- A 12-foot ladder leans against a building and reaches a window 10 feet above ground. What is the measure of the angle, to the nearest degree, that the ladder forms with the ground?
- In the diagram of equilateral triangle ABC shown below, E and F are the midpoints of AC and BC, respectively.

If EF = 2x + 8 and AB = 7x - 2, what is the perimeter of trapezoid ABFE? - Which information is not sufficient to prove that a parallelogram is a square?

(1) The diagonals are both congruent and perpendicular.

(2) The diagonals are congruent and one pair of adjacent sides are congruent.

(3) The diagonals are perpendicular and one pair of adjacent sides are congruent.

(4) The diagonals are perpendicular and one pair of adjacent sides are perpendicular.

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

25. Triangle A’B’C’ is the image of triangle ABC after a dilation with a scale factor of 1/2 and centered at point A. Is triangle ABC congruent to triangle A’B’C’? Explain your answer.

26. Determine and state the area of triangle PQR, whose vertices have coordinates P(-2,-5), Q(3,5), and R(6,1).

[The use of the set of axes below is optional.]

27. A support wire reaches from the top of a pole to a clamp on the ground. The pole is perpendicular to the level ground and the clamp is 10 feet from the base of the pole. The support wire makes a 68° angle with the ground. Find the length of the support wire to the nearest foot.

- In the diagram below, circle O has a radius of 10.
- On the set of axes below, triangle ABC ≅ triangle STU. Describe a sequence of rigid motions that maps triangle ABC onto triangle STU.
- In right triangle PRT, m∠P = 90°, altitude PQ is drawn to hypotenuse RT, RT = 17, and PR = 15.
- Given circle O with radius OA, use a compass and straightedge to construct an equilateral triangle inscribed in circle O. [Leave all construction marks.]

- Riley plotted A(-1,6), B(3,8), C(6,-1), and D(1,0) to form a quadrilateral.

Prove that Riley’s quadrilateral ABCD is a trapezoid.

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

Riley defines an isosceles trapezoid as a trapezoid with congruent diagonals. Use Riley’s definition to prove that ABCD is not an isosceles trapezoid. - A child-sized swimming pool can be modeled by a cylinder. The pool has a diameter of 6 1/2 feet and a height of 12 inches. The pool is filled with water to 2/3 of its height. Determine and state the volume of the water in the pool, to the nearest cubic foot.

One cubic foot equals 7.48 gallons of water. Determine and state, to the nearest gallon, the number of gallons of water in the pool.

- Nick wanted to determine the length of one blade of the windmill pictured below. He stood at a point on the ground 440 feet from the windmill’s base. Using surveyor’s tools, Nick measured the angle between the ground and the highest point reached by the top blade and found it was 38.8°. He also measured the angle between the ground and the lowest point of the top blade, and found it was 30°.

Determine and state a blade’s length, x, to the nearest foot.

- Given: Quadrilateral MATH, HM ≅ AT, HT ≅ AM, HE ⊥ MEA, and HA ⊥ AT

Prove: TA · HA HE · TH

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