Geometry Problems of the Day (Geometry Regents, August 2025 Part II)

This exam was adminstered in August 2025.

August 2025 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. Triangle D'A'N' is the image of △DAN after a translation. Explain why △D'A'N' must be congruent to △DAN.

Answer:

A translation is a rigid transformation that preserves size and shape. Therefore △D'A'N' must be congruent to △DAN.

That's all you need to write.

26. The table below lists five metals and their densities.

A solid metal cube has an edge length of 5 cm and a mass of 982.5 grams.

Using the table above, determine and state the type of metal from which this cube is made.

Answer:

Density is equalt to the mass divided by the Volume. Find the Volume, then find the density. Check your answer against the table.

V = (5)³ = 125 cm³.

d = 982.5 / 125 = 7.86

According to the table, the cube must be made of Iron.

27. The endpoints of CAS are C(–3,1) and S(7,6). Determine and state the coordinates of point A such that the ratio of CA:AS is 3:2.
[The use of the set of axes below is optional.]

Answer:

Graphing may make this easier to find. Or you can use a formula.

CA will be 3/5 of the length of CS, and AS will be 2/5 of the length of CS.

The difference of the x-coordinates is 10, and 3/5 (10) = 6. Add 6 to -3 and the x-coordinate of A is 3.

The difference of the y-coordinates is 5, and 3/5 (5) = 3. Add 3 to 1 and the x-coordinate of A is 4.

Point A is located at (3,4).

Another formula you use is the following:

(2/5)(-3,1) + (3/5)(7,6)
(-6/5, 2/5) + (21/5, 18/5)
(15/5, 20/5)
(3, 4)

I only started teaching this a year ago because the formula showed up in material. It looks counterintuitive, but it works. I showed it to my students. Many scratched their heads, but a few liked it and started using it.

28. The ramp shown in the diagram below has an angle of elevation of 4.8°. The ramp is built to a landing 0.6 m above the ground.

Determine and state the length of the ramp, to the nearest tenth of a meter.

Answer:

You have the height, which is opposite the angle, and you are looking for the hypotenuse. O and H means the SINE ratio.

sin (4.8°) = 0.6 / x

x = 0.6 / sin (4.8°) = 7.17...

The ramp has a length of about 7.2.

29. Angle KML is the vertex angle of isosceles triangle KLM below. Side LM is extended through vertex M to point N.

If m∠K = 15°, determine and state m∠KMN.

Answer:
KML is an isosceles triangle with M as the vertex angle. We are given the size of angle K, which means that we know the size of angle N.

Because of the Remote Angle Theorem, we know that m∠KMN = 15 + 15 = 30 degrees.

If you forgot the Remote Angle Theorem, you can add 15 and 15 to get 30, then subtract 180 - 30 to get 150, which is the size of angle KML. Since KML is supplementary to KMN, then KMN must be 180 - 150 = 30.

30. In the diagram below of circle L, the area of the shaded sector KLM is 7.5π and LK = 5.

Determine and state the degree measure of angle KLM, the central angle of the shaded sector.

Answer:
The Area of a sector is equal to the Area of the entire circle times the fraction of the circle that the sector represents. That fraction is the central angle divided by 360.

(x / 360) π(5)² = 7.5π

(x / 360) 25π = 7.5π

x = (360) (7.5π) / (25π)

x = 108

The measure of KLM is 108 degrees.

31. Using a compass and straightedge, construct the image of point A after a reflection over BC.
[Leave all construction marks.]

Answer: This is easier than drawing a perpendicular line. Make an arc with a length of BA, centered on point B. Make an arc with a length of CA, centered on Point C. The two arcs will meet at the reflection of point A.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

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:

25² - 20² = 625 - 400 = 225 = 15²

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.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

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.


9. On the set of axes below, △D'E'F' is the image of △DEF.

A transformation that maps △DEF onto △D'E'F' is a

(1) reflection over the line y = x
(2) reflection over the line y = - x
(3) point reflection through the origin
(4) translation 4 units left and 4 units down

Answer: (3) point reflection through the origin

A point reflection through the origin is the same as a 180 degree rotation. Choice (3) is the correct answer.

In Choice (1), a reflection over y = x would have the image in Quadrant I not III. Eliminate Choice (1).

In Choice (2), a reflection over y = -x would have E' closer to the x-axis and F' closer to the y-axis. The orientation is not correct for a reflection. Eliminate Choice (2).

In Choice (4), a translation doesn't change the orientation of the original shape, but D'E'F' has a different orientation. Eliminate Choice (4).

10. In circle O below, secants PCA and PDB are drawn from external point P.

If PA = 17, PD = 10, and BD = 12, what is the length of PC, to the nearest tenth?

(1) 7.1
(2) 7.7
(3) 12.9
(4) 14.2

Answer: (3) 12.9

The product of the length of the secant line times the length of the line that is outside the circle (PC, PD) will be equal for all of the secant lines.

That is to say: (PA)(PC) = (PB)(PD)

(17)(PC) = (10 + 12)(10)

17 (PC) = 220

PC = 220 / 17 = 12.94..., or 12.9

Choice (3) is the correct answer.

11. In the diagram below, CD || AB, and CB bisects ∠ABD.

Which statement must be true?

(1) CD ≅ AB
(2) AB ≅ BD
(3) △CDB is a right triangle
(4) △SNA is an isosceles triangle

Answer: (4) △SNA is an isosceles triangle

BC and BD are transversals crossing parallel lines. The means that ∠ABC ≅ ∠BCD. CB bisects ∠ABD, so ∠ABC ≅ ∠CBD. Since two angles in the triangle are congruent, then the triangle is an isosceles triangle.

Choice (4) is the correct answer.

12. Line h is represented by the equation y = 2/3 x - 4. Which equation represents the line that is perpendicular to line h and passes through the point (6,1)?

(1) y - 1 = 2/3(x - 6)
(2) y + 1 = 2/3(x + 6)
(3) y - 1 = -3/2(x - 6)
(4) y + 1 = -3/2(x + 6)

Answer: (3) y - 1 = -3/2(x - 6)

A line perpendicular to a line with a slope of 2/3 would have a slope of -3/2. Eliminate Choice (1) and (2).

Point slope form is y - h = m(x - k), with two minus signs. Those signs will flip to plus if the coordinate is negative. The slope of the line is m, and (h,k) is a point on that line.

Choice (3) is the correct answer.

Choice (4) goes through the point (-6,-1).

13. A wooden toy block can be modeled by a pyramid with a square base, as shown below. The height of the block is 17.4 cm and the square base has a side length of 8.2 cm.

The block is made of solid oak, which has a density of 0.77 g/cm³. What is the mass of the block, to the nearest gram?

(1) 300
(2) 506
(3) 637
(4) 901

Answer: (1) 300

The Volume of a pyramid is 1/3 the Area of the Base times the height. The Area of the square base is s². Density is mass divided by the Volume, so mass is equal to Density times Volume.

V = 1/3 (8.2)(8.2)(17.4) = 389.992

389.992 * 0.77 = 300.29..., which rounds to 300. Choice (1) is the correct answer.

If you forgot the (1/3), you would've gotten 901, which is Choice (4).

If you squared 17.4 instead of 8.2, you would've gotten 637, which is Choice (3).

If you found the correct Volume but divided by the Density, then you would've gotten 506, which is Choice (2).

14. In △ABC below, midsegment DE is drawn.

If DE = x + 3 and AC = 3x - 5, what is the length of DE?

(1) 28
(2) 14
(3) 7
(4) 4

Answer: (2) 14

Since DE is a midsegment, it is half the size of AC. That is, AC is twice the size of DE.

AC = 2 DE
3x - 5 = 2(x + 3)
3x - 5 = 2x + 6
x = 11

If x = 11, the DE = x + 3 = 11 + 3 = 14, which is Choice (2).

15. Triangle DUG is an isosceles right triangle with the right angle at G. If DU = 10√(2), what is the length of GU?

(1) 5
(2) 5√(2)
(3) 10
(4) 10√(2)

Answer: (3) 10

Side DU is the hypotenuse, not a leg. The two legs are congruent. The acute angles are each 45 degrees.

If you forgot the rules for Special Right Triangles, such as 45-45-90 (right isosceles), you can figure it out using Pythagorean Theorem.

x² + x² = (10√(2))²
2x² = 200
x² = 100
x = 10

Choice (3) is the correct answer.

16. In △RST below, RS = 9 cm, RT = 8 cm, and m∠TRS = 55°.

What is the area of △RST, to the nearest square centimeter?

(1) 59
(2) 36
(3) 29
(4) 21

Answer: (3) 29

Use the Law of Sines, A = 1/2 ab sin(C) to find the area, where a and b are sides of the triangle (not base and altitude) and C is the included angle between them.

A = 1/2 (8) (9) sin(55) = 29.4..., which rounds to 29.

Choice (3) is the correct answer.

More to come. Comments and questions welcome.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

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.


1. An equilateral triangle is continuously rotated around one of its altitudes. The three-dimensional object formed is a

(1) cone
(2) sphere
(3) cylinder
(4) pyramid

Answer: (1) cone

Imagine an equilateral triangle taped to a lollipop stick that running from one corner and through the midpoint of the opposite side. If you were to spin the stick, the stuff that the triangle will pass through will form a cone. That is Choice (1).

It is impossible to get any of the others from a triangle.

2. On the set of axes below, quadrilateral BDGF is rotated 90 degrees clockwise about the origin and then reflected over the y-axis. The image of quadrilateral BDGF is quadrilateral MQSP.

Side BD will always map onto

(1) MP
(2) PS
(3) MQ
(4) SQ

Answer: (3) MQ

Follow the movement. When BDGF is rotated 90 degree clockwise, the image appears in Quadrant IV with BD on top, closest to the x-axis. When this image is reflected across the y-axix, B'D' maps onto MQ, which is Choice (3).

3. In right triangle JOE, hypotenuse JO = 31.8 and m∠J = 38°. To the nearest tenth, the length of EJ is

(1) 19.6
(2) 25.1
(3) 40.4
(4) 51.7

Answer: (2) 25.1

Angle E is the right angle. EJ is adjacent to angle J, and you're given the hypotenuse. That means that you need to use cosine.

cos 38 = x / 31.8
x = 31.8 cos 38 = 25.0587... = 25.1

Choice (2) is the correct answer.

4. The hemisphere below has a radius of 8 cm.

To the nearest cubic centimeter, the volume of the hemisphere is

(1) 201
(2) 268
(3) 1072
(4) 2145

Answer: (3) 1072

Use the formula for the Volume of a Sphere and then take half of it. The formula is V = (4/3) π r³.

So V = (1/2) (4/3) π (8)³ = 1072.33...

Choice (3) is the correct choice.

If you'd forgotten the 1/2, you would've gotten choice (4).

If you'd forgotten the 1/2 AND halved the radius to 4, you would've gotten Choice (2).

If you used π r², you would've gotten Choice (1).
Yes, I needed to figure that one out. I wanted to know!

5. In parallelogram ABCD, diagonals AC and BD intersect at E. Which information is sufficient to prove ABCD is a rhombus?

(1) AE ≅ EC
(2) AC ≅ BD
(3) AB ⊥ BC
(4) AC ⊥ BD

Answer: (4) AC ⊥ BD

In a parallelogram, if the two diagonals are perpendicular, then the parallelogram is a rhombus.

6. Trapezoid JOSH, shown below, has non-parallel sides JH and OS, m∠J = 65°, m∠O = 30°, m∠OSA = 80°, and m∠SHU = 60°.

(1) 55°
(2) 60°
(3) 65°
(4) 70°

Answer: (4) 70°

SAO is a triangle, and the sum of its angles is 180 degrees. Because m∠O = 30° and m∠OSA = 80°, then m∠OAS = 70°. Angle OAS and angle HSA are alternate interior angles. Since JO || SH, then ∠OAS ≅ ∠OSA.

Therefore, m∠OSA = 70°, which is Choice (4).

7. In △ABC below, points D and E are on such that AC. DE || AC.

If AD = 8, DB = 4, and DE = 6, what is the length of AC

(1) 24
(2) 18
(3) 12
(4) 10

Answer: (2) 18

The two triangles are have sides that are proportional because they are similar triangles. If AD = 8 and DB = 4, then AB = 12, and the scale factor is 3.

Therefore, if DE = 6 then AC = 6 * 3 = 18.

Choice (2) is the correct answer.

8. On the set of axes below, circle C has a center with coordinates (2,–1).

Which equation represents circle C?

(1) (x - 2)² + (y + 1)² = 25
(2) (x - 2)² + (y + 1)² = 16
(3) (x + 2)² + (y - 1)² = 25
(4) (x + 2)² + (y - 1)² = 16

Answer: (1) (x - 2)² + (y + 1)² = 25

The formula for the equation of a circle is (x - h)² + (y - k)² = r², where (h,k) is the coordinates of the center and r is the length of the radius.

The center point C is at (1,-2), so eliminate Choices (3) and (4).

If you count across from C, you'll see that the radius is 5. (5)² = 25, so the correct answer is Choice (1).

More to come. Comments and questions welcome.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!