Geometry Problems of the Day (Geometry Regents, June 2026 Part II)

This exam was adminstered in June 2026.

June 2026 Geometry, Part II

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

25. On the set of axes below, △DAN ≅ △MIK.

Describe a sequence of rigid motions that maps △DAN onto △MIK.

Answer:

Make sure your transformations go in the correct direction, in this case from the top left to the lower right.

You should see that the orientation has changed in that the triangle is facing in the opposite direction. This indicates that a reflection has taken place. After that, a tranlation will move the figure to its image.

A reflection over the y-axis moves the triangle to Quadrant I with point A' at (1,8). Point I is at (1,-3) with is 11 units down and 0 units left or right.

The reflection over the y-axis (r_y-axis) is followed by a translation 11 units down. (T_0,-11).

This sequence could be done in the reverse order and achieved the same result.

26. In parallelogram LEAD below, C is on DE and S is on DA such that SC ⊥ DE.

If m∠LED = 21°, determine and state the measure of ∠CSA.

Answer:

Fill in what you know: LED = 21, so CDA must be 21 degrees as well, and angle DCS = 90 because it's a right angle.

This means that by the Remote Angle Theorem, angle CSA = 90 + 21 = 111 degrees.

If you forgot that theorem, you could have found that CSD = 180 - 90 - 21 = 69, and then CSA = 180 - 69 = 111.

27. A section of tree trunk from a white pine tree can be modeled by a cylinder. The diameter of the trunk is 1.5 feet, and the height of this section is 8 feet. If white pine weighs 25 pounds per cubic foot, determine and state the weight of this section of the white pine tree, to the nearest pound.

Answer:

You need to find the volume of the cylinder first. Once you have the volume, you can use that and the density to the mass (weight) using the formula m = VD.

The volume formula uses radius, but we were given diameter. Cut that in half. r = 1.5/2 = .75.

V = π r² h = π (7.5)² (8) = 14.137...

Leave the number in the calculator, or write down at least three decimal places. DO NOT ROUND IN THE MIDDLE OF A PROBLEM.

m = VD = 14.137 (25) = 353.429...

The weight is 353 to the nearest pound.

28. In △ADC below, points B and F are on AC and AD , respectively, such that AB = 40, BC = 30, and BF is drawn.

If AF = 64, determine and state the length of FD that would prove BF || CD.

Answer:

If BF || CD then triangles ABF and ACD are similar and their corresponding sides are proportional. Conversely, if the corresponding sides are proportional, then the triangles are similar and those two sides must be parallel.

Solve the proportion AB / BC = AF / FD

40 / 30 = 64 / FD

40 FD = (30)(64)

FD = (30)(64)/(40) = 48

It is also easy to see that QUAD could be broken up into four congruent triangles, for which the area could be found.

29. In △CLM below, m∠C = 33°, CL = 8, and CM = 15.

Determine and state, to the nearest tenth, the area of △CLM.

Answer:
You have two options here. The first is the use the Law of Sines: A = 1/2 (a)(b) sin C. The second is the use the sine function to find the height of the triangle, and then use A = 1/2 b h. This is the same exact thing but doing it in two steps might make it seem easier to you.

The Law of Sines allows you to find the area of a triangle without knowing the height if you know the lengths of two sides and the size of the angles included between them.

A = 1/2 (8) (15) sin (33) = 32.678...

The area is 32.7.

If you tried to find the height, you could use sin (33) = h / 8, and then h = 8 sin (33) = 4.357...

Then A = 1/2 (15)(4.357...) = 32.678

Which is the same answer.

30. In the diagram of △ABC below, use a compass and straightedge to construct the angle bisector of ∠ABC. [Leave all construction marks.]

Answer:
Starting at point B, make an arc that intersects the two sides of the angle. (I made a circle because it's easier with my paint program, but it isn't necessary.) Make two more arcs at the points where the first arcs met the sides. These points will intersect on the bisector of the angle. Draw a line from the intersection of these two arcs to the vertex of the angle.

31. The diagonals of parallelogram GRAM intersect at P. If RP = 12 and GA = 24, explain why GRAM is a rectangle.

Answer: The diagonals of a parallelogram biset each other. Therefore, if RP = 12 than RM = 24, which makes it congruent to GA.

If the diagonals of a parallelogram are congruent, then the parallelogram must be a rectangle.

End of Part II

How did you do?

Questions, comments and corrections welcome.
More to come.

Welcome to BURKE'S LORE Burke's Lore Briefs: Yesterday's Villains, the follow-up 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, June 2026 Part I)

This exam was adminstered in June 2026.

More Regents problems.

June 2026 Geometry Regents

Part I

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


1. Which transformation would result in the area of a rectangle's image being different from the area of its pre-image?

(1) a reflection over the y-axis
(2) a translation 4 units to the right
(3) a rotation of 90° counterclockwise about the origin
(4) a vertical stretch of scale factor 3 with respect to y = 0

Answer: (4) a vertical stretch of scale factor 3 with respect to y = 0

Reflections, translations and rotations are preserve the shape of the rectangle, and thus its Area (which is length times width).

A vertical stretch would change at least one of those dimensions.

The correct answer is Choice (4).

2. In the diagram below of /ABC, points D and E are the midpoints of CA and CB, respectively.

Which statement must always be true?

(1) DE = 1/2 AB
(2) DE = 1/2 AC
(3) AD = 1/2 AB
(4) AB = 1/2 DE

Answer: (1) DE = 1/2 AB

A midsegment of a triangle (a line connection the midpoints of two of the sides of a triangle) must be parallel to the third side and half of that sides length.

Choice (1) is the Correct answer. DE is half of the length of AB.

Choice (4) has this relationship in reverse. AB is twice the length of DE, not half.

In Choice (2), there is no relationship between the lengths of AC and DE. Eliminate Choice (2).

In Choice (3), there is no relationship between the lengths of AD and AB. Eliminate Choice (3).

3. In triangle RJM below, m∠R = 90° and MR = 7.25 inches.

If the measure of angle M is 35°, what is the length of MJ, to the nearest hundredth of an inch?

(1) 4.16
(2) 5.94
(3) 8.85
(4) 12.64

Answer: (3) 8.85

Since MJ is the hypotenuse of the triangle, it is the longest side. Therefore, Choices (1) and (2) can be eliminated.

The adjacent side to the angle is given, so we need to use the COS function.

Cos 35 = 7.25 / x
x Cos 35 = 7.25
x = 7.25 / Cos 35
x = 8.85

The correct answer is Choice (3).

4. A fish tank in the shape of a rectangular prism with a length of 35 cm, width of 17 cm, and a height of 25 cm is shown below.

If the fish tank is filled with water to a height 3 centimeters from the top, how many liters of water are in this tank, to the nearest liter?
[1 liter = 1000 cubic centimeters]

(1) 10
(2) 13
(3) 15
(4) 17

Answer: (2) 13

Volume equals length times width times height. However the height is reduced by three.

V = (35)(17)(22) = 13090

Convert to liters by dividing by 1000: 13.09 liters.

Choice (2) is the correct answer.

5. Triangle ACE is drawn below. Triangle ACE is mapped onto triangle AXE after a reflection over side AE. Triangle AXE is then mapped onto triangle LXE after a reflection over side XE.

Which side of △LXE is the image of AC?

(1) LE
(2) AX
(3) XE
(4) LX

Answer: (4) LX

Sketch the transformations in your booklet.

When reflecting over AE, AC maps onto AX.

When reflecting over XE, AE maps onto LX.

This is Choice (4).

6. The lines whose equations are represented by y = -1/2 x + 2 and x + 2y = 8 are

(1) parallel
(2) perpendicular
(3) the same line
(4) neither parallel nor perpendicular

Answer: (1) parallel

Check the slopes to see if they could be parallel or not.

The slope of the first line is -1/2.

To find the slope of the second line, rewrite the equation.

x + 2y = 8
2y = -x + 8
y = -1/2 x + 4

The slopes are the same but the y-intercepts are different. Therefore they are parallel and not the same line.

Choice (1) is the correct answer.

7. Right triangle ABC below has legs whose lengths are 6 and 8.

What is the volume of the three-dimensional object formed by continuously rotating △ABC about AB?
(1) 96π
(2) 128π
(3) 288π
(4) 384π

Answer: (2) 128π

Rotating about AB will create a cone with height AB (or 6) and radius BC (or 8).

V = 1/3 π (8)² (6) = 128π

Choice (2) is the Correct choice.

8. In △ABC below, CE is the perpendicular bisector of AEB.
Which statement is always true?

(1) AC ≅ BC
(2) AE ≅ CE
(3) ∠EAC ≅ ∠BCE
(4) m∠A + m∠B = 90°

Answer: (1) AC ≅ BC

Through SAS, we know that triangles CAE and CBE are congruent. Therefore AC ≅ BC, which is Choice (1).

In Choice (2), AE is congruent to BE, but not necessarily to CE. Eliminate Choice (2).

In Choice (3), angle EAC is congruent to angle EBC, but not to BCE. Eliminate Choice (3).

In Choice (4), angle A is congruent to angle B. They are not necessarily complementary unless they are both 45 degree angles. Eliminate Choice (4).

9. In right triangle ABC below, AB = 9, BC = 12, and altitude BD is drawn to hypotenuse AC.

Which equation is always true for BD?

(1) cos A = BD / 9
(2) sin C = BD / 12
(3) tan A = BD / 9
(4) sin C = BD / 15

Answer: (2) sin C = BD / 12

Know your rules for sin, cos, and tan, which are abbreviated SOH CAH TOA.

In Choice (1), cosine requires an adjacent side, but BD is opposite angle A. Eliminate Choice (1).

In Choice (2), sine requires the opposite side, which BD is, and the hypotenuse, which has a length of 12. This is the correct answer.

In Choice (3), tnagent requires the opposite and the adjacent, but 9 is the length of the hypotenuse in the triangle that has BD as a side. Eliminate Choice (3).

In Choice (4), 15 is the hypotenuse of triangle ABC, which doesn't have BD as a side. Eliminate Choice (4).

10. Which regular polygon, when rotated about its center, carries onto itself after both a 120° rotation and a 180° rotation?

(1) triangle
(2) square
(3) hexagon
(4) octagon

Answer: (3) hexagon

Which polygon is the same if rotated half of a complete rotation and one third of a complete rotation?

An equilateral triangle (not any triangle) is the same after 120 degrees, but not 180. Eliminate Choice (1).

A square is the same after 180 degrees, but not 120. Eliminate Choice (2).

A hexagon has six sides. It will be the same every 360/6 = 60 degree rotation. Both 120 and 180 are divisible by 60. This is the correct choice.

An octagon has eight sides. It will be the same every 360/8 = 45 degree rotation. However, 120 is not divisible by 45. Eliminate Choice (4).

h(-1) = (-1)² − 3(-1) + 1 = 1 + 3 + 1 = 5

The correction answer is Choice (3).

11. The coordinates of the endpoints of PA are P(3,-6) and A(-2, 9). If point C is on PA, what are the coordinates of C such that PC:CA = 1:4?

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

Answer: (4) (2, -3)

The ratio 1:4 means that CA is 4 times bigger than PC, and since 4 + 1 = 5, then C is 1/5 of the way from P to A.

The distance from 3 to -2 is -5, and 1/5 of -5 is -1. Subtract 1 from 3 to get 2, which is the x-coordinate of C. This eliminates all points except Choice (4), which is correct.

The distance from -6 to 9 is 15, and 1/5 of 15 is 3. Add 3 to -6 to get -3, which is the y-coordinate of C. This eliminates all points except Choice (4), which is correct.

12. On the set of axes below, △BLU is the image of △RED after a dilation.

What are the scale factor and the coordinates of the center of dilation of this transformation?

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

Answer: (4) 3 and (1,0)

Notice that point B is directly over point R. If the origin were the center of dilation, that wouldn't be possible. Both B and R must be directly above the center of dilation, which must be (from the choices) point (1,0). Eliminate Choices (1) and (3).

R is located at (1,3) and B is located at (1,9), which is three times farther away from (1,0). Therefore, the correct answer is Choice (4).

13. What are the coordinates of the center and the length of the radius of the circle whose equation is x² + y² = 45 + 4x?

(1) center (2, 0) and radius 7
(2) center (-2, 0) and radius 7
(3) center (2, 0) and radius 49
(4) center (-2, 0) and radius 49

Answer: (1) center (2, 0) and radius 7

Just looking at the choices, I "know" that the radius will be 7 and not 49, but "just knowing" is not how we do things here. (Usually.)

Rewrite the equation in standard form. You'll need to Complete the Square.

x² + y² = 45 + 4x

x² - 4x + y² = 45

x² - 4x + 4 + y² = 45 + 4

(x - 2)² + y² = 49

(x - 2)² + y² = 7²

The equation for a circle is (x - h)² + (y - k)² = r², where (h,k) is the center of the circle.

Since the minus sign is already in the formula, h is 2, not -2. And the radius is 7, which we now actually know.

Choice (1) is the correct answer.

14. In right triangle EFG below, altitude ET is drawn to hypotenuse FG.

If EF = 17 and FT = 15, what is the length of TG, to the nearest tenth?

(1) 3.8
(2) 4.3
(3) 8.0
(4) 9.1

Answer: (2) 4.3

There are two ways to calculate this.

First, the three right triangles you see are have corresponding sides that are proportional. Therefore, the ration of the long leg / hypotenuse will be the same. That means that FT / EF = EF / FG.

So 15 / 17 = 17 / (15 + TG)

15(15 + TG) = (17)(17)
225 + 15 TG = 289
15 TG = 64
TG = 64 / 15 = 4.266

TG = 4.3, which is Choice (2).

The second way to do this is to see that EFT is a right triangle, in the form x-15-17. You really, really, really should recognize that x = 8, but if you didn't recognize this Pythagorean Triple, you could use the Pythagorean Formula to get it.

Once you know that ET is 8, then you can use the Right Triangle Altitude Theorem, which says that FT/ET = ET/TG.

15/8 = 8/TG
15 TG = 64
TG = 64 / 15 = 4.266

We got the same answer.

15. In trapezoid ERJT, sides ER and TJ are parallel.
If m∠R = (2x + 15)° and m∠J = (3x- 40)°, what is m∠J?

(1) 125°
(2) 97°
(3) 83°
(4) 55°

Answer: (3) 83°

sides ER and TJ are parallel, then angles R and J must be supplementary because they are same-side interior angles on a transversal of parallel lines.

2x + 15 + 3x - 40 = 180
5x - 25 = 180
5x = 205
x = 41

The measure of angle J is 3(41) - 40 = 123 - 40 = 83 degrees, which is Choice (3).

16. In isosceles right triangle ABC, the length of hypotenuse AC is 14. What is the length of BC, to the nearest tenth?

(1) 7.0
(2) 8.1
(3) 9.9
(4) 19.8

Answer: (3) 9.9

In a right isosceles triangle, the length of the hypotenuse is the length of one leg times √(2).

√(2) s = 14
x = 14 / √(2) = 9.899...

Each leg is approximately 9.9 in length.

The correct choice is Choice (3).

17. In circle T below, tangent AS and secant ELS are drawn.

If SL = 8 and LE = 10, the length of AS is

(1) √(18)
(2) √(80)
(3) 9
(4) 12

Answer: (4) 12

The Tangent-Secant Rule is that the square of the tangent is equal to the product of the length of the secant times the length of the segment outside of the circle.

That is, (SA)² = (SL)(SE)
(SA)² = (8)(18) = 144
SA = 12

The correct choice is Choice (4).

18. Parallelogram MERT has diagonals that intersect at I. Which additional statement is sufficient to prove MERT is a rhombus?

(1) ∠ERT ≅ ∠RTM
(2) ∠MEI ≅ ∠RTI
(3) m∠TIM = 90°
(4) ME ≅ RT

Answer: (3) m∠TIM = 90°

The diagonals of a rhombus bisect each other and are perpendicular. The correct answer is Choice (3).

In Choice (1) if angles ERT and RTM are congruent, then MERT is a rectangle but not necessarily a rhombus.

In Choice (2) if angles MEI and RTI are congruent, then MERT is a parallelogram but not necessarily a rhombus.

In Choice (4) ME are RT are opposite sides which will always be congruent if in any parallelogram.

19. For the acute angles in right triangle ABC, sin(3x)° = cos(x + 10)°. What is the measure of the smallest angle in △ABC?

(1) 5°
(2) 15°
(3) 30°
(4) 60°

Answer: (3) 30°

The smallest angle in a right triangle cannot be 60 degrees. Eliminate Choice (4).

The sine of an angle is equal to the cosine of 90 minus that angle. So:

sin(3x)° = cos(x + 10)°
cos(90 - 3x)° = cos(x + 10)°
90 - 3x = x + 10
80 = 4x 20 = x

One angle is 3(20) = 60 and the other is (20) + 10 = 30, which is the smaller angle.

The correct choice is Choice (3).

20. Which two-dimensional shape below can not be a plane section of a rectangular prism?

(1) triangle
(2) octagon
(3) pentagon
(4) trapezoid

Answer: (2) octagon

The rectangular prism has six faces. If the plane section touches every side, the polygon with the most sides that can be created is a hexagon.

An octagon cannot be the result.

The correct choice is Choice (2).

21. Points A, B, and C are on circle D below such that DA = 12 and m∠ADB = 150°.
The length of AB is

(1) 5π
(2) 10π
(3) 24π
(4) 60π

Answer: (2) 10π

The length of the arc is 2πr times times the central angle / 360.

AB = (150 / 360) * (2) * (π) * (12) = 10π

The correct choice is Choice (2).

22. The lengths of two sides of a triangle are 12 and 30. The length of the third side could be

(1) 12
(2) 18
(3) 28
(4) 42

Answer: (3) 28

The Triangle Inequality Theorem says that the sum of the lengths of two sides of a triangle must be greater than the length of the third side.

The third side must be greater than 30 - 12 = 18 and less than 30 + 12 = 42.

The only choice with such a value is Choice (3).

23. In the diagram below, AC and ED intersect at B, and AE || CD.

If AB = 6, BC = 2, CD = 3, and BD = 4, what is the perimeter of △ABE?

(1) 9
(2) 18
(3) 21
(4) 27

Answer: (4) 27

The triangles are similar because of alternate interior angles and the vertical angle, and therefore the corresponding sides are proportional. Since AB corresponds to BC then we can calculate the scale factor of 6/2 = 3.

This means that because BCD has a perimeter of 9, then ABE must have a perimeter of (9)(3) = 27, which is Choice (4).

24. On the coordinate plane, a line is dilated by a scale factor, centered at a point on the line. The image of the line has

(1) the same slope and the same y-intercept as the original line
(2) the same y-intercept but a different slope as the original line
(3) the same slope but a different y-intercept as the original line
(4) a different slope and a different y-intercept than the original line

Answer: (1) the same slope and the same y-intercept as the original line

When a line is dilated and the center of dilation is a point on the line, the image will be the same line, which will by definition have the same slope and same y-intercept.

If the center of dilation is NOT on the line, the image would be a parallel line with the same slope by a different y-intercept.

This is Choice (1).

The other two choices are not possible with a dilation.

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!

Algebra Problems of the Day (Algebra Regents, June 2026 Parts III and IV)

This exam was adminstered in June 2026.

June 2026 Algebra, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

31. A ball is tossed up into the air from the deck of a building. The distance that the ball is above the ground t seconds after it is tossed can be modeled by the function D(t) = −16t² + 32t + 48, where the distance is measured in feet. On the set of axes below, graph D(t) = −16t² + 32t + 48.

State the maximum number of feet above the ground that the ball will reach.

State the number of seconds after the toss it will take the ball to hit the ground.

Answer:

Use your graphing calculator to find the points that you need to plot before t = 0 and when h(t) = 0.

You'll get the following points: (0,48), (1,64), (2,48), (3,0). Plot these and draw the curve. Do not draw any arrows at the endpoints.

The ball will reach a maximum height of 64 feet.

The ball will hit the ground at 3 seconds.

32. Solve the following system of equations algebraically for all values of x and y:

y = −2x + 3
y = x² − 5x + 3

Answer:

Set the expressions equal to each other and solve the quadratic equation.

x² − 5x + 3 = -2x + 3
x² − 3x = 0
x(x - 3) = 0
x = 0 or x - 3 = 0
x = 0 or x = 3

Solve for y:

If x = 0, then y = -2(0) + 3 = 3.

If x = 3, then y = -2(3) + 3 = -3

33. The table below shows the number of years of experience, x, working as a salesperson and their corresponding salaries, y, in thousands of dollars.

State the linear regression equation for these data. Round all values to the nearest hundredth.

State the correlation coefficient for this data set, to the nearest hundredth.

State what the correlation coefficient indicates about the linear fit of the data.

Answer:

Put the data into Lists L₁ and L₂. Make sure that "DiagnosticOn" has been executed. (It's in the Catalogue.) Run a Linear Regression.

COPY THE ENTIRE WINDOW OF DATA ONTO YOUR TEST PAPER. The reason for this is if you entered any piece of information incorrectly, all of the following answers are likely to be incorrect. However, if you show the data that you were given by your calculator, you can still earn three of the four points.

The equation for the linear equation will be y = 4.61x + 11.93

The correlation coefficient is r = 0.98.

The correlation coefficient indicates a strong correlation.

There isn't much to say about this. Either you know how to enter the data and run the linear regression, or you do not.

34. Graph the system of inequalities on the set of axes below.

2y < x − 8
3x + y ≥ 6

State the coordinates of a point that satisfies both inequalities. Justify your answer

Answer:

Rewrite the inequalities, solving for y in terms of x and graph.

Remember that < means a borken line that gets shading underneath, and ≥ means a solid line that gets shaded above.

2y < x − 8
y < 1/2 x − 4

3x + y ≥ 6
y ≥ -3x + 6

Your graph should look like this:

Any point in the area labeled "S" is a valid point that satisfies both inequalities. Do NOT choose any point on the solid line.

For example: (8, -2) is in the double-shaded area, marked "S".

June 2025 Algebra, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. At a department store in a tax-free state, Jane can either buy three tank tops and two sweatshirts for $52 or two tank tops and one sweatshirt for $30. If x represents the price of one tank top and y represents the cost of one sweatshirt, write a system of equations that could be used to model this situation. On the set of axes below, graph the system of equations.

State the coordinates of the point of intersection of your lines. Explain what each coordinate of the point of intersection means in the context of the problem.

Answer:

Write the equations from the first statement.

3x + 2y = 52 and 2x + y = 30

Rewrite these in slope-intercept form so you can graph them.

3x + 2y = 52
2y = -3x + 52
y = -3/2 x + 26

2x + y = 30
y = -2x + 30

Put these equations in your graphing calculator and plot the points from the tables of values.

Your graph will look like this.

The point of intersection is (8,14). In the context of the problem, tank tops cost $8 and sweatshirts cost $14.

End of Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! 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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If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra Problems of the Day (Algebra Regents, June 2026 Part II)

This exam was adminstered in June 2026.

June 2026 Algebra, Part II

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

25. Express the product of (1 − 2x) and (3 − 5x) as a polynomial in standard form.

Answer:

Use the Distributive Property, box method or even "FOIL". Just be sure to put it into standard form when combining like terms.

(1 − 2x) (3 − 5x)
= (1)(3 - 5x) - 2x(3 - 5x)
= 3 - 5x - 6x + 10x²
= 10x² - 11x + 3

26. On the set of axes below, graph f(x) = |x| − 3 over the domain −7 ≤ x ≤ 7.

Answer:

Use your graphing calculator. The abs() function is in the Catalogue.

An absolute value graph will be V shape. The vertex of this graph will be (0,-3). The slope on the left side is -1 and the slope on the right side is 1. The graph will be symmetrical.

Only graph between -7 and 7. The graph should have two endpoints. DO NOT put arrows on the graph. Do not go beyond -7 or 7. Follow instructions.

Your graph will look like this.

27. Solve the inequality algebraically: −4x + 1 > 9 + 3(2x + 1) + x

Answer:

Simplify each expression. Then isolate the variable with inverse operations.

−4x + 1 > 9 + 3(2x + 1) + x
−4x + 1 > 9 + 6x + 3 + x
−4x + 1 > 7x + 12
−11x > + 11
x < 1

28. Solve the formula A = 1/2 bh for h in terms of A and b.

Answer:

You can use a graphing calculator to give you a Five Number Summary of this data by putting it into a List.

A = 1/2 bh
2A = bh
2A / b = h

29. The table below shows the population of Manhattan for the years indicated, according to the U.S. Census Bureau.

Determine the average rate of change of the population per year between the years 1980 and 2020, rounded to the nearest integer.

Answer:
The average rate of change between the years 1980 and 2020 is just the slope of the line connecting those two ordered pairs.

(1,643,734 - 1,428,285) / (2020 - 1980) = 5386.225, which rounds to 5386.

30. Rewrite 5 / √(3) as a fraction with a rational denominator.

Answer:
Rationalize the denominator by multiplying by √(3) / √(3), which is equal to 1. (Multiplying a fraction by 1 doesn't change its value.)

5 / √(3) X √(3) / √(3) = 5√(3) / 3

End of Part II

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