This exam was adminstered in August 2025.
More Regents problems.
August 2025 Geometry Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
17. Triangle ABC is dilated by a scale factor of 2 to map onto its image,
△RST, on the set of axes below.
What are the coordinates of the center of this dilation?
(1) (1,-1)
(2) (2,1)
(3) (3,3)
(4) (0,0)
Answer: (2) (2,1)
Draw a line through SB, then draw one through RA, and one through TC. These three lines will meet at the center of dilation.
That point will be (2,1), which is Choice (2). Call this point O. OB has a length of √(5) and OS has a length of 2√(5)
18. What is the perimeter of △ABC, where the vertices have coordinates
A(–2,3), B(–2,–1), and C(6,–1)?
(1) 16
(2) 92
(3) 16√(5)
(4) 12 + 4√(5)
Answer: (4) 12 + 4√(5)
If you graph this on scratch paper (or even on question #17's graph), you will see that it's a right triangle.
AB has a length of 4, and BC has a length of 8, with is 12. The diagonal will be an irrational number, which cannot be combined with a rational number. Choice (4) is the only possible answer.
The square root of 42 + 82 is √(80) = √(16) √(5) = 4 √(5)
19. In the diagram below, GT and PF intersect at E, and ∠P ≅ ∠F.
equation is always true?
(1) PE/FE = FT/PG
(2) GE/TE = FT/PG
(3) PE/GE = TE/FE
(4) PE/FE = PG/FT
Answer: (4) PE/FE = PG/FT
Similar triangles have corresponding sides that are proportional. Find the equation that has the corresponding sides in the correct order.
In Choice (1), PE corresponds to FE and FT corresponds to PG but the second ratio is upside down. It compares small triangle over large triangle to large triangle over small triangle. Eliminate Choice (1).
In Choice (2), GE corresponds to TE and FT corresponds to PG but the second ratio is upside down. It compares small triangle over large triangle to large triangle over small triangle. Eliminate Choice (2).
In Choice (3), PE does not correspond to TE and GE does not correspond to FE. Eliminate Choice (3).
In Choice (4), PE corresponds to FE and PG corresponds to FT and they are in the correct order.
Choice (4) is the correct answer.
20. A section of sidewalk in the shape of a rectangular prism is being
replaced. The sidewalk is 10 feet long, 4 feet wide, and 4 inches deep.
A brand of concrete mix yields 0.6 cubic foot of concrete per bag.
What is the minimum number of bags of concrete mix that must be
purchased to completely replace this sidewalk?
(1) 22
(2) 23
(3) 26
(4) 27
Answer: (2) 23
Find the volume of the sidewalk. Remember that 4 inches is 1/3 of a foot. Then divide by 0.6. Then round UP. You cannot round down because you won't have enough concrete mix.
Number of bags = 10 * 4 * 1/3 / 0.6 = 22.222..., which rounds UP to 23.
Choice (2) is the correct answer.
21. The line 4x - 6y = 24 is transformed by a dilation of scale factor 3
centered at the origin. Which equation represents the image of the
line after this dilation?
(1) y = 2/3 x - 12
(2) y = 2/3 x - 4
(3) y = 2x - 12
(4) y = 2x - 4
Answer: (1) y = 2/3 x - 12
Since the choices are written in slope-intercept form, rewrite the equation into slope intercept form. Because the dilation has a scale factor of 3 centered on the origin, the y-intercept of the dilation will be three times the y-intercept of the original line.
4x - 6y = 24
-6y = -4x + 24
y = 4/6x - 24/6
y = 2/3 x - 4
This is the original line.
Therefore, the dilation will have an equation of y = 2/3 x - 12, which is Choice (1).
22. A rhombus is graphed on the set of axes below.
Which transformation does not carry the rhombus onto itself ?
(1) a rotation of 180° about the origin
(2) a rotation of 180° about point (1,0)
(3) a reflection over the line y = 1/2 x - 1/2
(4) a reflection over the line y = –2x + 2
Answer: (1) a rotation of 180° about the origin
This is a silly question because it's obvious that both Choices (1) and (2) cannot both be true, so one of them must be the answer.
A rhombus carries onto to itself if it is rotated 180 degrees around its center. The origin is not the center of this rhomus, so it will not carry the rhombus onto itself. Choice (1) is the correct answer.
Choices (3) and (4) reflect the rhombus over its diagonals. Both will carry the rhombus onto itself. Eliminate Choices (3) and (4).
23. In right triangle HAY below, altitude AL is drawn to hypotenuse HY.
If HY = 25 and YA = 20, the length of AL is
(1) 9
(2) 12
(3) 15
(4) 16
Answer: (2) 12
In triangle HAY, YA is the longer leg and HY is the hypotenuse. In triangle ALY, AL is the short leg and YA is the hypotenuse. This isn't helpful. However, in triangle HLA, AL is the longer leg and AH is the hypotenuse. We can calculate the length of HA using Pythagorean Thereom.
If you didn't immediately recognize that 15-20-25 is just a larger 3-4-5 triangle, do the math:
Write a proportion of corresponding side to find AL:
YA / HY = AL / AH
20 / 25 = AL / 15
25 AL = 300
AL = 12
Choice (2) is the correct answer.
24. Square ABCD has an area of 36. If the square is dilated by a scale factor of 1/2 centered at A, what is the area of its image?
(1) 9
(2) 18
(3) 72
(4) 144
Answer: (1) 9
If the size of a square is dilated by 1/2, then its perimeter will be 1/2 as big, but the area will be 1/4 of the original because both the length and the width are multiplied by 1/2, so the Area will be multiplied by 1/4.
One-quarter of 36 is 9, which is Choice (1).
The other way to think of it is that a square with an Area of 36 has a side length of 6. If the dilation is 1/2 the size, then it has a side length of 3. This would make the Area = 32 = 9.
End of Part 1. Comments and questions welcome.
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