Algebra 2 Problems of the Day (Algebra 2, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Algebra 2 Regents

Part I

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


17. If 4(10^5x-2) = 12, then x equals

(1) 2.3/5
(2) 1/3( log12/log40 + 5)
(3) (log(3) + 2) / 5
(4) 1/5 (log12/log4 + 2)

Answer: (3) (log(3) + 2) / 5

The rules for logarithms are derived from the rules of exponents. Remember that if you have difficulty. Use inverse operations to find x.

4(10^5x-2) = 12
10^5x-2) = 3
log₁₀3 = 5x - 2
log(3) + 2 = 5x
(log(3) + 2) / 5 = x

This is Choice (3).

18. A random sample of 152 students was surveyed on a particular day about how they got to school. The survey results are summarized in the table below.

Which statement is best supported by the data?

(1) The probability of being late given that a student walked is greater than the probability that a student walked given that the student was late.
(2) The probability of being late given that a student walked is less than the probability that a student walked given that the student was late.
(3) The probability of being late given that a student walked is equal to the probability that a student walked given that the student was late.
(4) The probability of being late given that a student walked cannot be determined.

Answer: (1) The probability of being late given that a student walked is greater than the probability that a student walked given that the student was late.

Write in the subtotals and the grand total to make the calculations easier:

In Choice (1), the probability of being late given that a student walked is 4/22. The probability that a student walked given that the student was late is 4/30. The first probability is greater than the second, so this is the correct answer.

Choice (2) says the first should be less, but it is not. Choice (3) says that the two are equal, but they are not. Choice (4) says the probability cannot be determined, but it can be. (We just did it!)

The correct answer is Choice (1).

19. If f(x) = ³√(x) + 4, then f^-1(x) equals

(1) ³√(x - 4)
(2) (x - 4)³
(3) x³ + 1/4
(4) -³√(x) - 4

Answer: (2) (x - 4)³

Substitute the x with a y, then set the expression equal to x. Finally, rewrite the equation to solve for y.

x = ³√(y) + 4
x - 4 = ³√(y)
(x - 4)³ = y

The correct answer is Choice (1).

20. Given the equation S(x) = 1.7sin(bx) + 12, where the period of S(x) is 12, what is the value of b?

(1) π/6
(2) 24π
(3) π/12
(4) 6π

Answer: (1) π/6

The period of a sine function is 2π/b. We want to find b. The "+ 12" in the function is meaningless to this problem because it has nothing to do with the period, which is also 12.

2π/b = 12
2π/12 = b
π/6 = b

Choice (1) is the correct answer.

21. Jin solved the equation √(4 - x) = x + 8 by squaring both sides. What extraneous solution did he find?

(1) -5
(2) -12
(3) 3
(4) 4

Answer: (2) -12

The extraneous value would be one that wouldn't make a valid equation. For examples, if a "solution" makes the expression x + 8 into a negative number, then that solution is extraneous because the value of the square root cannot be negative. The only possible answer from the four choices is (2) -12.

If you wanted to solve it to check (or if there had been more possibilities):

√(4 - x) = x + 8
4 - x = (x + 8)²
4 - x = x² + 16x + 64
0 = x² + 17x + 60
0 = (x + 12)(x + 5)
x + 12 = 0 or x + 5 = 0
x = -12 or x = -5

The solution x = -5 is valid because √(4 - (-5) = (-5) + 8 is a true statement becasue 3 = 3.

However, √(4 - (-12)) = (-12) + 8 is not a true statement because 4 =/= -4.

The correct answer is Choice (2).

22.The expression (x² + y²)² is not equivalent to

(1) (x² - y²)² + (2xy)²
(2) (x + y)⁴ + 2(xy)²
(3) x²(x²+2y²) + (y²)²
(4) (2x² + y²)² - (3x⁴ + 2x²y²)

Answer: (2) (x + y)⁴ + 2(xy)²

If you want a calculator "shortcut", replace x with a value such as 5 and y with a value such as 7 (note that 5 and 7 are mutually prime) and calculate all five expressions to see which doesn't match.

On first look, my "educated guess" would be Choice (2), but let's check all of them:

(x² + y²)² = x⁴ + 2x²y² + y⁴

In Choice (1) (x² - y²)² + (2xy)²
= x⁴ - 2x²y² + y⁴ + (4xy)²
= x⁴ + 2x²y² + y⁴

In Choice (2), (x + y)⁴ is NOT equal to x⁴ + y⁴.
(x + y)⁴ + 2(xy)²
= x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ + 4x²y²
This is totally different from the original expression. Choice (2) is the correct answer.

In Choice (3) x²(x²+2y²) + (y²)²
= x⁴ + 2x²y² + y⁴

In Choice (4) (2x² + y²)² - (3x⁴ + 2x²y²)
= 4x⁴ + 4x²y² + y⁴ - 3x⁴ - 2x²y²
= x⁴ + 2x²y² + y⁴

23. The height of a running trail is modeled by the quartic function y = f(x) shown below, where x is the distance in miles from the start of the trail and y is the height in feet relative to sea level.
If this trail has a minimum height of 16 feet below sea level, which function(s) could represent a running trail whose minimum height is half of the minimum height of the original trail?

I. y = f(1/2 x); II. y = f(x) + 8; III. y = 1/2f(x)

(1) I, only
(2) II, only
(3) I and III
(4) II and III

Answer: (4) II and III

Half of the minimum height would be 8 feet below sea level instead of 16 feet. That mean that the minimum is either raised by 1/2 or raised by 8 feet.

In function I, putting 1/2 in front of x inside the parentheses, the function has changed the period but hasn't changed either the baseline or the amplitude. Function I is no good, so you can eliminate Choices (1) and (3).

Function II must be true because it's in both of the remaining choices. Adding 8 to the y value will raise the minimum from -16 to -8.

Function III shrinks the y value by 1/2, so -16 will become -8. Function III fits the description.

Choice (4) is the correct answer.

24. The crew aboard a small fishing boat caught 350 pounds of fish on Monday. From that Monday through the end of the week on Friday, the weight of the fish caught increased 15% per day. The total weight, in pounds, of fish caught is approximately

(1) 411
(2) 612
(3) 1748
(4) 2360

Answer: (4) 2360

Use the formula for the Geometric Series to find the sum:

S₅ = (350 - 350(1.15)⁵) / (1 - 1.15)

P = 2359.833...

The correct answer is Choice (4).

End of Part I. Comments and questions 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!

Algebra 2 Problems of the Day (Algebra 2, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Algebra 2 Regents

Part I

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


9. In a small city, there are 22 gas stations. The mean price for a gallon of regular gas was $2.12 with a standard deviation of $0.05. The distribution of the data was approximately normal. Given this information, the middle 95% of the gas stations in this small city likely charge

(1) $1.90 to $2.34 for a gallon of gas
(2) $1.97 to $2.27 for a gallon of gas
(3) $2.02 to $2.22 for a gallon of gas
(4) $2.07 to $2.17 for a gallon of gas

Answer: (3) $2.02 to $2.22 for a gallon of gas

The middle 95% would be within 2 stadard deviations from the mean, which would be + 10 cents from the mean price.

That gives 2.12 - .10 = 2.02 and 2.12 + .10 = 2.22. This is Choice (3).

10. The expression (4x² - 5) / (x² - 1) is equivalent to

(1) 4 - 1 / (x² - 1)
(2) 4 + 1 / (x² - 1)
(3) 4 - 9 / (x² - 1)
(4) 4 - 4 / (x² - 1)

Answer: (1) 4 - 1 / (x² - 1)

This is an easy one to factor. Four times x² - 1 is 4x² - 4. Therefore...

(4x² - 5) / (x² - 1) =
( (4x² - 4) - 1) / (x² - 1) =
(4x² - 4) / (x² - 1) - 1 / (x² - 1) =
4 - 1 / (x² - 1)

Not to be flippant about this, but this is the kind of division/factoring that can be done in my head. However, I still write it down because you never know when you'll accidentally flip a sign.

The correct answer is Choice (1).

11. For all positive values of x, which expression is equivalent to √(x) * ⁴√(x¹¹)?

(1) x^19/22
(2) x^11/8
(3) x^13/4
(4) x^2/11

Answer: (3) x^13/4

Use the laws of exponents to multiply by adding the fractions.

√(x) * ⁴√(x¹¹) =
x^1/2 * x^11/4 =
x^{2/4 + 11/4} =
x^13/4

The correct answer is Choice (3).

12.The expression i²(5x - 2i)² is equivalent to

(1) -25x² + 20xi - 4
(2) -25x² + 20xi + 4
(3) 25x² + 20xi + 4
(4) 25x² + 4

Answer: (2) -25x² + 20xi + 4

Rewrite i² as -1 and then square the binomial.

i²(5x - 2i)² =
-1(5x - 2i)² =
-1( 25x² - 10xi - 10xi + 4i^2/) =
-1( 25x² - 20xi - 4) =
-25x² + 20xi + 4 =

Watch out for -- and double check! -- all your signs!

Choice (2) has the x-values and only the x-values, and this is the correct answer.

13. Functions f and g are given below.

f(x) = 7/2 x² - 5x + 11
g(x) = 3x² - 7x + 25
When 2f(x) is subtracted grom g(x), the result is

(1) 4x² - 3x - 3
(2) -4x² + 3x + 3
(3) 4x² - 17x + -47
(4) -4x² - 17x + 47

Answer: (2) -4x² + 3x + 3

Substitute the expressions into g(x) - 2f(x). The "from" goes first!

3x² - 7x + 25 - 2(7/2 x² - 5x + 11) =
3x² - 7x + 25 - 7x² + 10x - 22 =
-4x² + 3x + 3

This is Choice (2).

14. A manufacturer claims that the number of ounces of a beverage dispensed by one of its automatic dispensers is normally distributed with a mean of 8.0 ounces and a standard deviation of 0.04 ounces. To the nearest tenth of a percent, what percent of the cups filled by this company’s dispenser will contain between 7.9 and 8.11 ounces?

(1) 99.5
(2) 99.4
(3) 99.1
(4) 97.6

Answer: 3) 99.1

Use your graphing calculator for this. Enter the following command: normCdf(7.9,8.11,8,0.04)

The first two terms as the lower and upper limits of the range that you are looking at. The third term is the mean. The fourth term is the standard deviation.

The output will be 0.9908, which is 99.08%, which rounds to 99.1%.

I'm glad we don't have to calculate this by hand!

15. What is the value of x in the solution of the system of equations below?

5x + 2y - z = 14
7y - z = 31
5y + 4z - 5x = -23

(1) -17
(2) 2
(3) -1/5
(4) -7

Answer: (4) -7

They went out of their way to make this one confusing! You should rewrite the system, lining up the like terms first. You can solve any system of two equations with two variables by eliminating (or subsituting) one of the variables to get one equation with one variable. You can solve a system of three equations with three variables by eliminating one variable, to that you will have two equations with two variables, and then continue as above.

5x + 2y - z = -14
7y - z = 31
-5x + 5y + 4z = -23

From the middle equation, we have x = 7y - 31, which can be substituted into the first and third equations.

5x + 2y - 7y + 31 = -14
-5x + 5y + 28y - 124 = -23

5x - 5y = -45
-5x + 33y = 101

28y = 56
y = 2

Now solve for x:
5x - 5(2) = -45
5x - 10 = -45
5x = -35
x = -7

This correct answer is Choice (4).

If you wanted to solve for z, you would do this:
7(3) - z = 31
21 - z = 31
z = 21 - 31 = -10

Double-check your work. I messed up a sign the first time through. Luckily, I didn't get one of the other choices!

16. The graph below shows the amount of a radioactive substance left over time.

The daily rate of decay over an 8-day interval is approximately

(1) 23%
(2) 95%
(3) 5%
(4) 77%

Answer: (3) 5%

The substance dropped from 50 to about 38 the first day, and from 38 to 30 on the second day. Let's start with that.

(50 - 38)/50 = .24 rate of decay.

(38 - 30)/38 = .21 rate of decay.

(30 - 23)/30 = .23 rate of decay.

You can see that the daily decay rate is about 23%, which is Choice (1).

None of the other choices are even close.

If you calculate 50(1 - 0.23)⁸, you'll get a result about 6.2, which aligns with the graph for Day 8.
More to come. Comments and questions 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!

Algebra 2 Problems of the Day (Algebra 2, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Algebra 2 Regents

Part I

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


1. The exact value of sin(8π/3) is

(1) 1/2
(2) -1/2
(3) -√(3)/2
(4) √(3)/2

Answer: (4) √(3)/2

Consider the unit circle, which has a length of 2π. The fraction 8/3 is equal to 2 and 2/3, so 8π/3 is the same as 2π/3, which is Quadrant II where sin is negative. Eliminate Choices (1) and (4).

The point on the unit circle is (-1/2, √(3)/2), where cos is the x value and sin is the y value.

The correct answer is Choice (4).

2. A teacher randomly divides all of her students into two groups. She grades the homework for one group but does not grade the homework for the other group. All homework is returned to the students. She then compares test scores for each of the groups to see if grading homework has an effect on the test scores.
This method of data collection is best described as

(1) an experiment
(2) an unbiased survey
(3) a simulation
(4) an observational study

Answer: (1) an experiment

This is an experiment. It isn't a survey or a simulation and it isn't observational in nature.

The correct answer is Choice (1).

3. Which expression is equivalent to (x - 2)² + 27(x - 2)² - 90?

(1) (x + 30)(x - 3)
(2) (x + 28)(x - 5)
(3) (x - 30)(x + 3)
(4) (x - 2)(x + 25)(x - 90)

Answer: (2) (x + 28)(x - 5) down

Replace (x - 2) with another variable, such as y, and factor the polynomial:

y² + 27y² - 90
(y + 30)(y - 3)
(x - 2 + 30)(x - 2 - 3)
(x + 28)(x - 5)

The correct answer is Choice (2).

4. Given the functions f(x) = 2x + 5/2 and g(x) = 3/x, what are the solutions to f(x) = g(x)?

(1) (0.75, 4) or (-2,-1.5)
(2) x = 0.75 or x = -2
(3) y = -1.5 or y = 4
(4) (-2, 0.75)

Answer: (2) x = 0.75 or x = -2

Note that all four choices have the same values, so you don't have to solve anything.

Finding solutions to f(x) = g(x) means "for what values of x are the two functions equal?" That means that you are looking for the x values, not the function values.

Choice (2) has the x-values and only the x-values, and this is the correct answer.

5. Given f(x) = 2x³ - 3x² - 5x - 12 and g(x) = x - 3, the quotient of f(x)/g(x) is

(1) 2x² + 3x + 4
(2) 2x³ + 3x² + 4x
(3) 2x² - 9x + 22 - 78/(x - 3)
(4) 2x³ - 9x² + 22x - 78

Answer: (1) 2x² + 3x + 4

If 2x³ is divided by (x - 3), the highest power in the quotient will be 2, so eliminate Choices (2) and (4).

Use polynomial long division to solve this:

This is Choice (1).

6. Abby is told that each day there is a 50% chance it will rain. Which simulation can Abby perform to determine the likelihood of it raining for the next seven days?

(1) Flip a coin seven times, count how many heads, and repeat 50 times.
(2) Roll a die seven times, count how many twos, and repeat 50 times.
(3) Roll a pair of dice, count totals of seven, and repeat 50 times
(4) Flip a coin 50 times and count how many heads.

Answer: (1) Flip a coin seven times, count how many heads, and repeat 50 times.

Running a simulation means running the test multiple times and analyzing the results. Choice (4) doesn't do that.

In Choice (1), there is a 50% chance that the coin will land heads, and the simulation is repeated 50 times. This is the correct answer.

In Choice (2), there is only a 1/6 probability of rolling a two, not 50%. Eliminate Choice (2).

In Choice (3), there is only a 1/6 probability of rolling a seven on two dice. Eliminate Choice (3).

7. What are the solutions to 4x² - 7x - 2 = -10?

(1) -1/4, 2
(2) 7/8 + √(79)/8 i
(3) 7/8 + √(241)/8
(4) 7/8 + √(143)/8 i

Answer: (2) 7/8 + √(79)/8 i

Set the equation equal to 0 and use the Quadratic Formula.

4x² - 7x + 8 = 0

x = ( -(-7) + √((-7)² - 4(4)(8) ) / (2(4))

x = ( 7 + √( 49 - 128 ) / (8))

x = ( 7 + √( -79 ) / (8))

x = 7/8 + i √(79)/8

This is Choice (2).

8. If x - 5 is a factor of p(x) = ax⁴ + bx³ + cx² + dx + e, then which statement must be true?

(1) p(-5) = 0
(2) p(-5) =/= 0
(3) p(5) = 0
(4) p(5) =/= 0

Answer: (3) p(5) = 0

If x - 5 is a factor of p(x), then by definition, p(5) = 0. This is Choice (3).

We don't know anything about p(-5).

More to come. Comments and questions 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, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Geometry Regents

Part I

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


17. A glass fish tank is designed to be placed on a stand in the corner of a room with perpendicular walls. The tank can be modeled using part of a cylinder, as shown below. The inner length of the fish tank along the wall is 22 inches, and the height of the tank is 27 inches.

How much water, to the nearest gallon, does the fish tank hold? [1 gal = 231 in³]

(1) 44
(2) 59
(3) 89
(4) 178

Answer: (1) 44

The sides are prependicular because it fits into the corner of a room. That means that there is a 90 degree angle and this tank is 1/4 of a complete cylinder.

So volume = 1/4 π r² h
V = 1/4 π (22)² (27)
V = 3267π
V = 10263.583 in³

10263.583 * 1 gal / 231 in³ = 44.43...

The correct answer is Choice (1).

18. Line m, whose equation is y = -2x + 8, is dilated by a scale factor of 1/2 centered at the origin. Which equation represents the image of line m?

(1) y = -x + 4
(2) y = -2x + 4
(3) y = -x + 8
(4) y = -2x + 8

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

The dilation will not affect the slope of the line so Choices (1) and (3) can be eliminated.

Since the dilation is centered on the origin, the y-intercept of the image will be half the distance of the original y-intercept. Therefore, the y-intercept of the image is +4. This is Choice (2).

19.In right triangle RST below, altitude SV is drawn to hypotenuse RT.

Which statement is always true?

(1) RT/ST = ST/VT
(2) VR/VT = VT/VS
(3) RV/SV = SV/RT
(4) TR/VR = VR/SR

Answer: (1) RT/ST = ST/VT

There are three right triangles in this image and all three of them are similar, which means that hte corresponding sides are proportional. Look for the proportion that lists corresponding sides.

In Choice (1), the hypotenuse and the long leg of the biggest triangle are compared to hypotenuse and the long leg of the right side triangle. This is the correct answer.

In Choice (2), the first ratio compares two sides that are not in the same triangle while the second ratio has two sides in the right triangle. Eliminate Choice (2).

In Choice (3), the first ratio contains the short leg and long leg of the left triangle and the second ratio has the short leg of the right triangle and the hypotenuse of the big triangle. Eliminate Choice (3).

In Choice (4), the hypotenuse of the big triangle is over a leg of a different triangle. Eliminate Choice (4).

20. What is the measure, in radians, of a central angle that intercepts an arc length of 12π cm in a circle with a diameter of 36 cm?

(1) π/6
(2) π/3
(3) 2π/3
(4) 3π/2

Answer: (3) 2π/3

The diameter is 36 cm, so the radius is 18 cm. Divide the arc length of 12π by 18 and you'll get 12π/18, which simplifies to 2π/3.

This is Choice (3).

21.A regular nonagon has a center point, P. What degree of rotation about point P will carry the nonagon onto itself?

(1) 60°
(2) 90°
(3) 180°
(4) 200°

Answer: (4) 200°

Every ninth of a full rotation will map a nonagon onto itself because a nongon has nine sides.

Because 360/9 = 40, the correct choice must be a multiple of 40 degrees. The only possible answer is Choice (4).

22. If two sides of a triangle have lengths of 2 and 7, the length of the third side could be

(1) 9
(2) 8
(3) 5
(4) 4

Answer: (2) 8

The sum of the lengths of the two shortest sides of a triangle must be greater than the length of the longest side. Another way to say it is that the sum of any two sides always must be greater than the length of the third side.

If two sides are 7 and 2, then the third side must be less than 7 + 2 or 9 and greater than 7 - 2 or 5.

The third side CANNOT be 4, 5, or 9. Only 8 is a possible side length for this triangle.

Note that 5 and 9 would create overlapping line segments, not triangles. If the smaller sides are 2 and 4, then they couldn't intersect because the endpoints are too far away from each other.

The correct answer is Choice (2).

23. The car tire shown in the photograph below has a diameter of 2 1/4 feet.

Approximately how many rotations will the tire make in one mile?

(1) 373
(2) 747
(3) 1328
(4) 2347

Answer: (2) 747

Find the circumference of the tire, in feet, and then divide 5280 feet by that number.

C = (3.141592...)(2.25) = 7.068582...

5280 / 7.068582 = 746.967355

It would take 747 rotations to travel one mile, which is Choice (2).

I want to make a airplane joke here...

24. In quadrilateral TOWN, OW ≅ TN and OT ≅ WN. Which additional information is sufficient to prove quadrilateral TOWN is a rhombus?

(1) ON ⊥ TW
(2) TO ⊥ OW
(3) OW || TN
(4) ON and TW bisect each other.

Answer: (1) ON ⊥ TW

The properties of a rhombus are that the opposite sides are parallel, the four sides are congruent, and the diagonals are perpendicular. The sides are not perpendicular unless the rhombus is also a square.

In Choice (1), ON and TW are diagonals, so you need them to be perpendiuclar. This is the correct answer.

In Choice (2), TO and OW are consecutive sides. They are only perpendicular in squares, but do not have to be perpendicular otherwise. Eliminate Choice (2).

In Choice (3), OW and TN must be parallel but this is not sufficient to prove the shape is a rhombus. It could be a trapezoid. Eliminate Choice (3).

In Choice (4), the diagonals of a rhombus only bisest each other if the shape is also a square (because this is a property of rectangles, not rhombuses). Eliminate Choice (4).

End of Part I. Comments and questions 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, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Geometry Regents

Part I

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


9. Scalene triangle JKL is drawn below.

If median LM is drawn to side KJ, which statement is always true?

(1) LM = KM
(2) KM = 1/2 KJ
(3) LM ⊥ KJ
(4) ∠KLM ≅ ∠JLM

Answer: (2) KM = 1/2 KJ

The median connects angle L with the midpoint of KJ. Therefore, KM is half of the length of KJ.

LM is not congruent to KM because LM is the median. KM is congruent to JM, but they are each half of KJ.

10. In circle O, chord KA intersects diameter YN at S.

If mYK = 120° and mYA = 105°, what is m∠ASN?

(1) 22.5°
(2) 75°
(3) 97.5°
(4) 120°

Answer: (3) 97.5°

Since YN is a diameter, we can find the measures of the other two arcs. Once we know those, the measure of angle ASN will be the average of the measures of arcs YK and AN.

If YK = 120 degrees, then KN = 60 degrees. If YA = 105 degrees, then AN = 75 degrees.

The angle ASN will be the average, half the sum, of the measures of arcs YK and AN. So 1/2(120 + 75) = 1/2(195) = 97.5, which is Choice (3).

11.Triangle ABC is graphed on the set of axes below. The vertices of △ABC have coordinates A(-3,4), B(-5,-1), and C(3,-2).

What is the area of △ABC?

(1) 16
(2) 20
(3) 21
(4) 24

Answer: (3) 21

The easiest way to find the area of a triangle that does line up with the grid is to create a box around it and then find the area of the three triangles that you cut away from the rectangle.

If you make a box from point C(3,-2) to (3,4) to (-5,4) to (-5,-2), you have a rectangle with an area of 8 x 6 = 48 square units. The three extraneous triangles will have areas of 1/2(2)(5) = 5, 1/2(1)(8) = 4, and 1/2(6)(6) = 18. Subtract 48 - 5 - 4 - 18 = 21, which is Choice (3).

In this example, it would've been possible to divide the triangle along the line y = -1. In that case, the top triangle would have an area of 1/2(7)(5) = 17.5, and the bottom triangle would have an area of 1/2(7)(1) = 3.5. The sum of 17.5 + 3.5 = 21.

12. In △ABC below, DE is a midsegment, and BD ≅ DE.
Which statement is always true?

(1) △ABC is isosceles
(2) △ABC is scalene
(3) BD ≅ BE
(4) DA ≅ EC

Answer: △ABC is isosceles

If BD ≅ DE, then BA ≅ AC because the latter two line segments are twice the size of the first two line segments. If you double the size of congruent segments, you will get another pair of congruent segments that are twice the size.

That means that ABC is an isosceles triangle because two sides, BA and AC, are congruent.

Even though the image shown appears to be scalene, there is no reason that the triangle described in the question must be scalene. You cannot assume that the image shown is drawn to scale and covers all possibilities.

There is nothing indicating that BD must be congruent with BE. Moreover, if BD were congruent to BE, then DA would have to be congruent to EC as well. They can't both be true.

13. As shown in the diagram below, JKL || MNOP, KRN, and OR ≅ ON.

If m∠POR = 116°, what is m∠LKN?

(1) 58°
(2) 116°
(3) 122°
(4) 128°

Answer: (4) 128°

Work your way through the angles you know and the ones you can work out.

If m∠POR = 116°,then m∠NOR = 64° because they are a linear pair. If m∠NOR = 64°, then m∠NRO = 64° because NOR is an isosceles triangle and OR ≅ ON. Then m∠ONR = 52° because 180 - 64 - 64 = 52 degrees.

Since JKL || MNOP, then ∠LKN and ∠PNK are supplementary because they are same-side interior angles. Therefore m∠LKN = 180 - 52 = 128, which is Choice (4).

14. The ratio of similarity of square ABCD to square WXYZ is 2:5. If AB = x + 3 and WX = 3x + 5, then the perimeter of ABCD is

(1) 8
(2) 20
(3) 32
(4) 80

Answer: (3) 32

Solve for x using the lengths of AB and WX and the ratio between them. Once you know the length of one side, you can find the perimeter of the square.

5(x + 3) = 2(3x + 5)
5x + 15 = 6x + 10
5 = x

AB = 5 + 3 = 8
Perimeter of ABCD is 4 * 8 = 32.

The correct answer is Choice (3).

15. In parallelogram ABCD below, diagonals AC and BD intersect at E.

Which transformation would map △ABC onto △CDA?

(1) a reflection over AC
(2) a reflection over DB
(3) a clockwise rotation of 90° about point E
(4) a clockwise rotation of 180° about point E

Answer: (4) a clockwise rotation of 180° about point E

A reflection over the diagonals would not line up correctly because neither AC nor BD are angle bisectors. The answer must be a rotation.

To move A to C, the triangle would have to be rotated 180 degrees about point E. That's Choice (4), which is the correct answer.

16. The square pyramid drawn below has a volume of 175.

If the height of the pyramid is 21, what is the perimeter of the base?

(1) 5
(2) 10
(3) 20
(4) 25

Answer: (3) 20

If the Volume is 175 and the height is 21, then the area of the base can be found using the Volume formula:

V = 1/3 (Area of Base) (height)
175 = 1/3 (A) (21)
175 = 7 A
25 = A

If the Area of the square is 25, then the length of one side is the square root of 25, which is 5. If the length of one side is 5, then the Perimeter of the square base is 4 * 5 = 20, which is CHoice (3).

More to come. Comments and questions 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.
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Geometry Problems of the Day (Geometry Regents, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Geometry Regents

Part I

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


1. On the set of axes below, △AB'C' is the image of △ABC.

What is the scale factor and center of dilation that maps △ABC onto △AB'C'?

(1) 1/2 and the origin
(2) 2 and the origin
(3) 1/2 and vertex A
(4) 2 and vertex A

Answer: (4) 2 and vertex A

Point A does not move, so it is the center of dilation. B' is twice as far away from A as point B is, so the scale factor is 2.

2. Line segment PAQ has endpoints whose coordinates are P(-2,6) and Q(3,-4). What are the coordinates of point A, such that PA:AQ = 2:3?

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

Answer: (4) (0,2)

It may help to sketch this or use the graph paper in the back of the booklet.

To get from P to Q, the x-coordinate increases by 5 and the y-coordinate decreases by 10.

Two-fifths of 5 is 2, and two-fifths of -10 is -4. So point A is 2 units to the right of P and 4 units down. That puts it at (0,2).

Another way to solve this is to use a formula:

(3/5)(-2,6) + (2/5)(3,-4)
(-6/5, 18/5) + (6/5, -8/5)
(0,10/5)
(0,2)

It looks crazy, but it works.

3. On the set of axes below, congruent parallelograms ABCD and RSTU are graphed.

Which sequence of transformations maps ABCD onto RSTU?

(1) a reflection over the x-axis followed by a translation ten units to the left and one unit up
(2) a translation four units down followed by a reflection over the y-axis
(3) a reflection over the y-axis followed by a translation of two units down
(4) a translation ten units to the left followed by a reflection over the x-axis

Answer: (2) a translation four units down followed by a reflection over the y-axis down

The orientation has changed, so it is not a translation. And from the new direction, we can see that it is a reflection and not a rotation of any kind.

Translating four units down puts A' at (2,0), B' at (8,-1), etc. Reflecting A'B'C'D' over the y-axis brings it to RSTU. Choice (2) is correct.

Choice (1) is incorrect because the image wouldn't match up. A wouldn't transform to R, B to S, etc.

4. Triangle ABC has a right angle at C. If AC = 7.7 and m∠B = 24°, what is AB, to the nearest tenth?

(1) 18.9
(2) 17.3
(3) 8.4
(4) 3.1

Answer: (1) 18.9

Triangle ABC has right angle C, which means that leg AC is across from angle B. You could sketch this confuses you at all.

AB is the hypotenuse of the triangle, so you are supposed to use the sine function to solve this problem.

Before we do that, however, we can eliminate 3.1, because it's not the longest side of the triangle. Second, since angle B is only 24 degrees, and angle A is therefore 66 degrees, then BC must be much bigger than 7.7 and definitely bigger than 8.4. So we've eliminated two choices.

If I were to "bet" (as opposed to "guess" and we shouldn't do either), I'd think that (1) will be the answer.

Sin 24 = 7.7 / x, so x = 7.7 / sin 24 degrees = 18.93..., which is Choice (1).

If you used Tangent, you would've gotten Choice (2), and if you'd used Cosine, you would've gotten Choice (3).

5. Given △PQR and △LMN with PQ ≅ LM, which additional statement is sufficient to always prove △PQR ≅ △LMN?

(1) QR ≅ MN and ∠R ≅ ∠N
(2) QR ≅ MN and ∠Q ≅ ∠M
(3) QR ≅ MN and ∠P ≅ ∠L
(4) QR ≅ MN and ∠P ≅ ∠M

Answer: (2) QR ≅ MN and ∠Q ≅ ∠M

Because we were given a pair of congruent sides and because all the choices have a pair of congruent sides and a pair of congruent angles, then we could prove the triangles are congruent using SAS. That means that we need angles that are included (that is, "between") the corresponding sides.

If we have PQ and QR corresponding to LM and MN, respectively, then the included angles must be Q and M. That is Choice (2).

6. The equation of a circle is x² + 6y = 4x - y² + 12. What are the coordinates of the center and the length of the radius?

(1) center (2,-3) and radius 5
(2) center (-2,3) and radius 5
(3) center (2,-3) and radius 25
(4) center (-2,3) and radius 25

Answer: (1) center (2,-3) and radius 5

First, get the equation into the correct form by moving everything to the left side of the equal sign, leaving the 12 on the right, and then Complete the Squares.

x² + 6y = 4x - y² + 12
x² - 4x + y² + 6y = 12
x² - 4x + 4 + y² + 6y + 9 = 12 + 4 + 9
(x - 2)² + (y + 3)² = 25
(x - 2)² + (y + 3)² = 5²

The correct answer is Choice (1) center (2,-3) and radius 5.

7. A square with a side length of 3 is continuously rotated about one of its sides. The resulting three-dimensional object is a

(1) cube with a volume of 9.
(2) cube with a volume of 27.
(3) cylinder with a volume of 27π.
(4) cylinder with a volume of 54π.

Answer: (3) cylinder with a volume of 27π.

If you spin a square around, you will get a cylinder. You cannot get a cube from this circular motion.

The cylinder will have a radius of 3 and a height of 3, so the Volume will be πr²h, or π(3)²(3), which is 27π. The correct answer is Choice (3).

8. Line k is represented by the equation 4y + 3 = 7x. Which equation represents a line that is perpendicular to line k and passes through the point (-5,2)?

(1) y + 2 = 4/7 (x - 5)
(2) y - 2 = 4/7 (x + 5)
(3) y + 2 = -4/7 (x - 5)
(4) y - 2 = -4/7 (x + 5)

Answer: (3) y + 2 = -4/7 (x - 5)

First, if the point (-5,2) is on the line than we can elimate Choices (1) and (3), which go through point (5,-2).

Second the original line has a slope of 7/4, which you get when you divide both sides by 7. The inverse reciprocal of that is -4/7, which is the slope of the perpendicular line. That eliminates Choice (1) and (2), so only Choice (3) remains, which is the Correct answer.
More to come. Comments and questions 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 older 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!