Geometry Problems of the Day (Geometry Regents, June 2025 Part I)

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Geometry Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


9. A candle can be modeled by a pyramid with a square base, as shown below. The height of the candle is 10 cm, and each side of the base measures 6 cm.

If the candle wax burns at a rate of 3.5 cubic centimeters per hour, what is the approximate number of hours this candle could burn?

(1) 103
(2) 51
(3) 34
(4) 11

Answer: (3) 54

A two-part problem because first you need to know how many cubic centimeters are in the candle before you can answer the question.

The volume of a square pyramid is 1/3 (side)² times height, which is (1/3) (6)² (10) = 120 cm³.

If it burns at a rate of 3.5 cm³/hr, then divide 120 / 3.5 = 34.2857..., or about 34.

Choice (3) is the correct answer.

If you forgot the (1/3), you would've gotten choice (1). Oops.

10. In the diagram below, tangent SR and secant STU are drawn to circle O from external point S.

If TU ≅ RU and mTR = 68°, what is m∠S?

(1) 22°
(2) 34°
(3) 39°
(4) 78°

Answer: (3) 39°

The measure of angle S is one half of the difference of arc RU and arc TR. You need to find the size of RU.

Since TU ≅ RU, and TU + RU + TR = 360, then 2(RU) + 68 = 360, and 2(RU) = 292, and RU = 146.

One half of the difference of RU and TR is 1/2(146 - 68) = 39, which is Choice (3).

11. Triangle RST has m∠S = 33°, RS = 7, and ST = 12. What is the area of △RST, to the nearest tenth?

(1) 22.9
(2) 27.3
(3) 35.2
(4) 45.7

Answer: (1) 22.9

Use the Law of Sines: A = 1/2 ab sin C = (1/2)(7)(12)sin(33) = 22.87...

12. Triangle ABC, with vertices whose coordinates are A(–3,–2), B(–1,2), and C(4,1), is graphed on the set of axes below.

Triangle A'B'C', whose vertices have coordinates A'(–6,–2), B'(–2,2), and C'(8,1), is the image of nABC. The transformation that maps △ABC onto △A'B'C' is a

(1) dilation
(2) translation
(3) vertical stretch
(4) horizontal stretch

Answer: (4) horizontal stretch

It is not a dilation because the y-coordinates don't change. On the other hand, the x-coordinates of the image are double the original.

Choice (4) is the correct answer.

13. Which equation represents a line parallel to the line represented by y = 4x + 6 and passing through the point (–3,2)?

(1) y - 2 = 4(x + 3)
(2) y + 3 = 4(x - 2)
(3) y - 2 = -1/4(x + 3)
(4) y + 3 = -1/4(x - 2)

Answer: (1) y - 2 = 4(x + 3)

Parallel lines have the same slope so the answer must have a slope of 4. Eliminate choices (3) and (4).

The answers are given in point-slope form. The x-coordinate of the given point goes with the x variable, and the y-coordinate goes with the y variable.

Choice (1) is the correct answer.

14. Which two-dimensional figure is always formed when a plane intersects a right cylinder perpendicular to its base?

(1) circle
(2) triangle
(3) rhombus
(4) rectangle

Answer: (4) rectangle

If you slice a round cake down the center and separate the two pieces, you will see two rectangles, composed of the top, bottom and the sides of the cake.

Choice (4) is the correct answer.

15. In △KMP below, CE is drawn parallel to MP.

If KC = 8, CM = 3, and CE = 12, what is the length of MP?

(1) 24
(2) 16.5
(3) 15
(4) 4.5

Answer: (2) 16.5

The sides of the two triangles are proportional, but first you need to find KM = KC + CM = 8 + 3 = 11.

So 8/12 = 11/MP.

Then MP = 11 * 12 / 8 = 16.5

The correct answer is Choice (2).

16. A parallelogram must be a rectangle if its diagonals

(1) are perpendicular
(2) bisect each other
(3) bisect its angles
(4) are congruent

Answer: (4) are congruent

What do you know about parallelograms and rectangles?

The diagonals of parallelograms and rectangles are not perpendicular unless they are also rhombuses. Eliminate Choice (1).

The diagonals of both parallelograms and rectangles are bisect each other, so that is not enough information. Eliminate Choice (2).

The diagonals of parallelograms and rectangles only are bisect each other is they are also rhombuses. Eliminate Choice (3).

The diagonals of parallelograms are only congruent if the parallelogram is a rectangle. Choice (4) is the answer.

More to come. Comments and questions welcome.

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