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.


17. Point O divides COA such that CO:OA = 1:4. If C has coordinates (–2, –9) and A has coordinates (3,6), the coordinates of O are

(1) (2,3)
(2) (1,0)
(3) (0,-3)
(4) (-1,-6)

Answer: (4) (-1,-6)

Dividing 1:4 means that there are 5 parts, with CO being 1/5 of the length COA and OA being 4/5 of the length. Find 1/5 of the change in x values and 1/5 of the change of y values and add those values to (-2,-9).

1/5(3 - -2) = 1/5(5) = 1. 1/5(6 - -9) = 3.

(-2 + 1, -9 + 3) = (-1, -6), which is choice (4).

Another formula you can use is this:

4/5(-2,-9) + 1/5(3,6)
(-8/5, -36/5) + (3/5, 6/5)
(-5/5, -30/5)
(-1, -6)

Remember to put the 4/5 in front of the first coordinate and the 1/5 in front of the second.

This always works.

18. A spherical balloon is fully inflated with helium to a diameter of 1.7 feet. If helium costs $0.80 per cubic foot, what is the cost to completely fill the balloon with helium?

(1) $2.06
(2) $2.42
(3) $3.22
(4) $16.46

Answer: (1) $2.06

Find the volume of the sphere and then multiply by 0.80.

The volume of a sphere is V = 4/3 π r³

V = 4/3 (3.141592...)(.85)³ = 2.57244...

Multiply that by $0.80 = 2.0579..., which rounds to $2.06.

11. In right triangle ABD below, altitude BC is drawn to hypotenuse AD, AC = 2.5, and CD = 4.3.

What is the length of BA, to the nearest tenth?

(1) 3.3
(2) 3.4
(3) 4.1
(4) 5.4

Answer: (3) 4.1

The three right triangles are all similar and their sides are proportional. That means that AC / AB = AB / AD (short leg / hypotenuse).

2.5 / x = x / (2.5 + 4.3)
x² = 17
x = 4.123...

This rounds to 4.1, which is Choice (3).

20. Trapezoid ZOYD has parallel sides ZO and DY. If m∠Z = 141° and m∠Y = 73°, what is m/D?

(1) 39°
(2) 73°
(3) 107°
(4) 141°

Answer: (1) 39°

If ZO and DY are parallel sides, then OY and DZ are the other two sides. The sum of the measures of angles O and Y must be 180 degrees, and the sum of the measures of angles D and Z must be 180 degrees.

180 - 141 = 39.

Choice (1) is the correct answer.

21. Triangle ABC is translated 5 units to the left and 2 units up to map onto △PQR. Which statement is not always true?

(1) △PQR ≅ △ABC
(2) ∠A ≅ ∠Q
(3) BQ = √(29)
(4) RQ = CB

Answer: (2) ∠A ≅ ∠Q

In a translation, size and shape are preserved. The corresponding angles will be congruent. However, in ABC and PQR, angles A and Q are NOT corresponding angles. They do not have to be congruent. They could be if the triangle is isosceles or equilateral.

In Choice (1), the triangles are always congruent. Eliminate Choice (1).

In Choice (3), point B moves 5 left and 2 up to point Q, so line segment BQ would be √(5^2 + 2^2), which is √(29). Eliminate Choice (3).

In Choice (4), RQ corresponds to CB, so length is preserved. Eliminate Choice (4).

22. In the diagram below, congruent triangles PEN and PAL are drawn.

Which rigid motion maps nPEN onto nPAL?

(1) a point reflection of △PEN through P
(2) a reflection of △PEN over the angle bisector of ∠EPA
(3) a rotation of △PEN about point P, mapping PE onto PA
(4) a translation of △PEN along EA, mapping point E onto A

Answer: (2) a reflection of △PEN over the angle bisector of ∠EPA

The triangles do not have the same orientation. There is a reflection involved.

A point reflection would be 180 degrees about point P. Eliminate Choice (1).

A reflection over the a line bisect the image will carry PEN onto PAL. Choice (2) is the correct answer.

A rotation mapping PE to PA would not move N to point L. The result would be a kite. Eliminate Choice (3).

A translation would not change the orientation. Moving E to A would create a bowtie-like image. Eliminate Choice (4).

23. A cone has a height of 8 inches and volume of 75.4 cubic inches. What is the diameter of the cone, to the nearest inch?

(1) 9
(2) 2
(3) 3
(4) 6

Answer: (4) 6

Use the formula for the volume of a cone, substitute what you know, and solve for the unknown.

The formula for the volume of a cone is V = 1/3 π r² h.

V = 1/3 π r² h
75.4 = 1/3 π r² (8)
r² = (3) 75.4 / (8 π)
r² = 9.00...
r = 3

So the DIAMETER is 2 * 3 = 6, which is Choice (4).

24. The line represented by the equation 5x - 2y = 10 is transformed by a dilation centered at (2,0) with a scale factor of 2. The image of the line

(1) is the original line
(2) passes through the point (4,0)
(3) passes through the point (0,–10)
(4) is perpendicular to the original line

Answer: (4) are congruent

A dilation of a straight line will be a parallel line or the same line if the center of dilation is on the line itself. The image cannot be have a different slope, so it cannot be perpendicular. Eliminate Choice (4).

Substitute (2,0) into the original question: 5(2) - 2(0) = 10. This is a true statement, so (2,0) is a point on the original line. Therefore, the image will be the same line. Choice (1) is the correct answer.

End of Part I. Comments and questions welcome.

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