Algebra Problems of the Day (Algebra Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Algebra Regents

Part I

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


9. The zeros of the function f(x) = x(x - 5)(3x + 6)

(1) 0, -5, and 2
(2) 0, 5, -2
(3) -5 and 2, only
(4) 5 and -2, only

Answer: (2) 0, 5, -2

The zeroes are the values x that make the function equal to zero.

Since the first of the three terms that are being multiplied is just x, then the funtion is equal to 0 when x = 0, because zero times anything is zero. Eliminate Choices (3) and (4).

The second term (x - 5) is equal to 0 when x = 5 because (5 - 5) = 0. Eliminate Choice (1).

Choice (2) is the answer because (5 - 5) = 0 and (3(-2) + 6) = 0.

10. What is the y-intercept of the line that passes through the points (-1,5) and (2,-1)?

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

Answer: (3) 3

The y-intercept is the value of y when x = 0, or the y-coordinate when the line crosses the y-axis.

Before you do anything, it should be obvious that the number must be in-between 5 and -1, so eliminate Choices (1), (2), and (4).

Choice (3) is the only possible Choice and is the correct answer.

If you didn't realize this, you can use the scratch graph paper to plot the points and draw the line, using a straightedge or ruler, to find the answer.

You could also find the slope of the line: (-1 - 5) / (2 - -1) = -6 / 3 = -2.

If the slope is -2, then the point after (-1, 5) is (0, 3), which is Choice (3).

11. Nancy has just been hired for her first job. Her company gives her four choices for how she can collect her annual salary over the first eight years of employment.
Each function below represents the four choices she has for her annual salary in thousands of dollars, where t represents the number of years after she is hired.

a(t) = 2^t + 25
b(t) = 10t + 75
c(t) = √(400t) + 80
d(t) = 2(t + 1)² - 10t + 50
Which pay plan should Nancy choose in order to have the highest salary in her eighth year?

(1) a(t)
(2) b(t)
(3) c(t)
(4) d(t)

Answer: (1) a(t)

First, an important note to all high school students taking this exam: THIS WILL NEVER HAPPEN IN REAL LIFE!

They are asking which will be the highest amount a(8), b(8), c(8) or d(8). That's it. Forget about salaries and jobs. Just put them in the calculator, and be sure that you entered them correctly when it comes to the exponents and square roots.

a(8) = 2⁸ + 25 = 281

b(8) = 10(8) + 75 = 155

c(8) = √(400(8)) + 80 = 136.57

d(8) = 2(8 + 1)² - 10(8) + 50 = 132

The maximum amount is Choice (1), a(t), which you might have suspected because it is the only one that is exponential, and exponential functions rise more quickly than other functions. But it doesn't hurt to check.

12. The third term in a sequence is 25 and the fifth term is 625. Which number could be the common ratio of the sequence?

(1) 1/5
(2) 5
(3) 1/25
(4) 25

Answer: (2) 5

The sequence is increasing, so the common ratio must be greater than 1. Eliminate Choices (1) and (3).

If you multiply, 25 * 5 = 125 and 125 * 5 = 625, you have the correct answer. Choice (2) is the correct choice.

If you multiply 25 * 25 = 625, then the third term is 625, not the fourth term.

How you're supposed to solve this, if you didn't have choices to work with, is as follows:

25 * x is term 3, and term 3 * x = 625, so 25 x² = 625.
That means that x² = 25, and x = 5.

13. The box plot below summarizes the data for the amount of snowfall, in inches, during the winter of 2021 for 12 locations in western New York.

What is the interquartile range?

(1) 30
(2) 50
(3) 80
(4) 110

Answer: (2) 50

The interquartile range is the difference between the third or upper quartile (Q3) and first or lower quartile (Q1). Basically, it's the length of the box.

Q3 = 110, and Q1 = 60, and 110 - 60 = 50, so the IQR = 50, which is Choice (2).

Choice (1), 30, is the length from the median to Q3. Eliminated Choice (1).

Choice (3), 80, is the median of the data, not the length of the box. Eliminate Choice (3).

Choice (4), 110, is Q3, not the interquartile range. Eliminate Choice (4).

14. Four quadratic functions are represented below.

Which function has the smallest minimum value?

(1) I
(2) II
(3) III
(4) IV

Answer: (1) I

You can check the minimum value of function c(x) in your calculator, or you can work it out.

Function a(x) is in vertex form, meaning that (3,-7) is the vertex, so the minimum value for a(x) = -7. The minimum value of b(x) is -4 and the minumum value of d(x) is -5. Eliminate Choices (2) and (4).

The minimum value of c(c) is on the Axis of Symmetry, which is the line with the equation x = -b/(2a) = -(6)/(2) = -3. The value of c(-3) = (-3)² + 6(-3) + 3 = -6. Eliminate Choice (3).

Choice (1), I, is the correct answer.

15. The equation that represents the sequence -2, -5, -8, -11, -14, . . . is

(1) aⁿ = -3 + (-2)(n - 1)
(2) aⁿ = -2 + (-3)(n - 1)
(3) aⁿ = 3 + (-2)(n - 1)
(4) aⁿ = -2 + (3)(n - 1)

Answer: (2) aⁿ = -2 + (-3)(n - 1)

The initial term is -2 and the common difference is -3. The Choice that shows this is Choice (2), which is the Correct answer.

Check: Let's let n = 4 to find the value of the fourth term

-3 + (-2)(4 -1) = -9, not -11. Eliminate Choice (1).
-2 + (-3)(4 - 1) = -11. This is correct.
3 + (-2)(4 - 1) = -3, not -11. Eliminate Choice (3).
-2 + (3)(4 - 1) = 7, not -11. Eliminate Choice (4).

Only Choice (2) gives the correct fourth term.

16. The dot plot below shows the number of goals Jessica scored in each lacrosse game last season.

Which statement about the dot plot is correct?

(1) mean > mode
(2) mean = median
(3) mode = mediam
(4) median %gt; mean

Answer: (2) mean = median

The mode is 3, because there are 5 of them. The median is 2.5 because there are 20 pieces of data and the 10th is 2 and the 11th is 3, so the median is 2.5. The median does not equal the mode, so eliminate Choice (3).

The other three choices use the mean, so you have to calculat the mean, which is the average of all the data. That is, add up all 20 numbers and divide by 20. You can take a shortcut by using multiplication for each individual score. Obviously, I don't need to add the zeros, but I'll write it down just for completeness.

(3 * 0) + (3 * 1) + (4 * 2) + (5 * 3) + (2 * 4) + (2 * 5) + (1 * 6) = 50
50 / 20 = 2.5

The mean is not greater than the mode, so Choice (1) is eliminated.

The mean is 2.5 which is equal to the median. The correct answer is Choice (2).

More to come. Comments and questions welcome.

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Algebra Problems of the Day (Algebra Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Algebra Regents

Part I

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


1. A ball was launched into the air, and its height above the ground was recorded each second, as shown in the table below.

Based on these data, which statement is a valid conclusion?

(1) The ball lands on the ground at 4 seconds.
(2) The ball reaches a maximum height of 11 feet.
(3) The ball was launched from a height of 0 feet.
(4) The ball reaches its maximum height at 2 seconds.

Answer: (4) The ball reaches its maximum height at 2 seconds.

The ball starts at 11 feet from the ground, which is at 0 feet. Read the table to find the correct answer.

Choice (1) says the ball is on the ground at 4 seconds. It is not. It is at 11 feet. Eliminate Choice (1).

Choice (2) says the ball is reaches a maximum of 11 feet. No, it starts at 11 feet and goes up to 75 feet. Eliminate Choice (2).

Choice (3) says the ball was launched from 0 feet. No, it starts at 11 feet at 0 seconds. Eliminate Choice (3).

Choice (4) says the ball reaches its maximum height at 75 feet. The point (2,75) is the highest in the table. Also, the numbers on either side are symmetrical, so you can be sure that (2,75) is the vertex of the parabola that these points would graph. This is the correct answer.

2. A tour bus can seat, at most, 48 passengers. An adult ticket costs $18 and a child ticket costs $12. The bus company must collect at least $650 to make a profit. If a represents the number of adult tickets sold and c represents the number of child tickets sold, which system of inequalities models this situation if they make a profit?

(1) a + c < 48 ; 18a + 12c > 650
(2) a + c < 48 ; 18a + 12c > 650
(3) a + c < 48 ; 18a + 12c < 650
(4) a + c < 48 ; 18a + 12c < 650

Answer: (2) a + c < 48 ; 18a + 12c > 650

At most means "is less than or equal to". At least means "is greater than or equal to".

Immediately eliminate Choice (1) and (3) which don't have the "equal to" portion.

Choice (2) has the correct symbols that match the language in the question.

3. Which equation is always true?

(1) x² • x³ = x⁵
(2) 3^x • 3² = 9^2x
(3) -z² = z²
(4) 7^a • 7^b = 7^ab

Answer: (1) x² • x³ = x⁵

When you multiply two terms that have the same base, you keep the base and add the exponents.

In Choice (1), x² • x³ = x x • x x x = x⁵. This is the correct answer.

In Choice (2), 3^x • 3² = 3^(x+2), not 9^2x. Eliminate Choice (2).

In Choice (3), -z² = z² is not true for any value other that z = 0 because of the Order of Operations. What is true is (-z)² = z². Eliminate Choice (3).

In Choice (4), 7^a • 7^b = 7^a+b , not 7^ab. Eliminate Choice (4).

4. The expression -2(x² - 2x + 1) + (3x² + 3x - 5) is equivalent to

(1) x² + x - 4
(2) x² - x - 7
(3) x² + 7x - 4
(4) x² + 7x - 7

Answer: (4) x² + 7x - 7

Use the distributive property and then combine like terms.

-2(x² - 2x + 1) + (3x² + 3x - 5)
-2x² + 4x - 2 + 3x² + 3x - 5
x² + 7x - 7

5. Which sum is irrational?

(1) -2√(12) + √(100)
(2) -√(4) + 1/3 √(900)
(3) 1/2 √(25) + √(64)
(4) √(49) + 3√(121)

Answer: (1) -2√(12) + √(100)

You could put each of these into your calculator and see if you get an infinite, non-repeating decimal as answer. If the numbers under the radicals are perfect squares, then the numbers are rational numbers. The sum of two rational numbers is always rational. The sum of a rational and an irrational number is also irrational. (The sum of two irrational numbers is usually, but not always, irrational.)

Looking at the choices above, 12 is not a perfect square, but 100, 4, 900, 25, 64, 49, and 121 are perfect squares.

Choice (1) is the only choice that adds an irrational number to a rational number and is therefore irrational. This is the correct answer.

Choice (2) equals -2 + 10 = 8.

Choice (3) equals 2.5 + 8 = 10.5

Choice (4) equals 7 + 33 = 40.

6. The solution to (4(x - 5)) / 3 + 2 = 14 is

(1) 15
(2) 14
(3) 6
(4) 4

Answer: (2) 14

You can substitute each answer in your calcluator if you aren't sure how to solve this.

Use inverse operations to isolate the variable and solve for x:

(4(x - 5)) / 3 + 2 = 14
(4(x - 5)) / 3 = 12
4(x - 5) = 36
x - 5 = 9
x = 14

Check: (4(14 - 5)) / 3 + 2 = (4(9)) / 3 + 2 = (36) / 3 + 2 = 12 + 2 = 14. Check!

The correct answer Choice (2).

7. On an island, a rare breed of rabbit doubled its population each month for two years. Which type of function best models the increase in population at the end of two years?

(1) linear growth
(2) linear decay
(3) exponential growth
(4) exponential decay

Answer: (3) exponential growth

Continously doubling is growth not decay. The rate of change is not constant, so it is not linear.

This is an example of exponential growth. This is a definition question. There is nothing to solve. You either understand the concept or you do not.

8. What is the degree of the polynomial 2x - x² + 4x³?

(1) 1
(2) 2
(3) 3
(4) 4

Answer: (3) 3

This is another concept/definition question. The degree of a polynomial is the value of the highest exponent. This is the term that would go first when written in standard form. The correct answer is 3, which is Choice (3).

Choice (1), 1, is the value of the exponent of the first term that is written, but it isn't the highest exponent. Eliminate Choice (1).

Choice (2), 2, is the coefficient of the first term that is written, not the exponent. It's the exponent of the second term, but that doesn't matter. Eliminate Choice (2).

Choice (4), 4, is the leading coefficient, meaning that it would be the coefficient of the first term if the expression was written in the correct order (standard form). Eliminate Choice (4)

More to come. Comments and questions welcome.

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

June 2024 Algebra Regents Part IV

This exam was adminstered in January 2024 .

June 2024 Algebra, Part IV

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

35. Courtney went to a coffee shop to purchase lattes and donuts for her friends. One day she spent a total of $15.50 on four lattes and two donuts. The next day she spent a total of $18.10 on three lattes and five donuts. All prices included tax.

If x represents the cost of one latte and y represents the cost of one donut, write a system of equations that can be used to model this situation.

Courtney thinks that one latte costs $2.75 and one donut costs $2.25.
Is Courtney correct? Justify your answer.

Use your equations to determine algebraically the exact cost of one latte and the exact cost of one donut.

Answer:

Answer every section of this question to get full credit. if you make a mistake on top, you will not be penalized more than once if you are consistent until the end and give a final answer. Make sure you use x and y and not L and D because the questions says to use x and y!

You should be able to read the first two sentences and translate them into equations.

4x + 2y = 15.50
3x + 5y = 18.10

Two find out if Courtney is correct, you only have to substitute those values into BOTH equations to see if those values make BOTH equations true.

4(2.75) + 2(2.25) = 15.50 (check)
3(2.75) + 5(2.25) = 19.50 =/= 18.10 (does not check!)

Courtney is incorrect.

Finally, copy your system of equations and solve through elimination. Note that it says algebraically, so you won't get full credit for graphing the equations in your calculator and finding the intersection point.

4x + 2y = 15.50
3x + 5y = 18.10

12x + 6y = 46.50
12x + 20y = 72.40

14y = 25.90

y = $1.85

The price of one donut is $1.85. If you got an repeating decimal, or anything that doesn't come out to an exact number of dollars and cents, that should tell you that there was an error somewhere. The problem deals with money and must have a monetary answer.

Now that you know the price of a donut, you can find the price of a latte by using either of the original equations:

4x + 2y = 15.50
4x + 2(1.85) = 15.50
4x + 3.70 = 15.50
4x = 11.80
x = 2.95.

The price of a latte is $2.95.

End of exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

June 2024 Algebra Regents Part III

This exam was adminstered in January 2024 .

June 2024 Algebra, Part III

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

31. Graph the following system of equations on the set of axes below.

y = x² - 3x - 6
y = x - 1
State the coordinates of all solutions.

Answer:

Use your calculator and check the table of values to find the points. Only label the points of intersection, and be sure to label at least one line.

Your graph should look like this one:

The solutions are the points of intersection: (-1,-2) and (5,4)

32. The table below shows the amount of money a popular movie earned, in millions of dollars, during its first six weeks in theaters.

Write the linear regression equation for this data set, rounding all values to the nearest hundredth.

State the correlation coefficient to the nearest hundredth.

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

Answer:

For a linear regression, put all of the data into two lists on your graphing calculator and run the regression. Make sure you have "DiagonsticOn" in the Catalog.

The equation will be y = -37.57x + 215.67

The correlation coefficient is -.98.

This means that there is a strong negative correlation between the weeks and the dollars earned.

33. Use the quadratic formula to solve the equation 3x² - 10x + 5 = 0. Express the answer in simplest radical form.

Answer:

You must use the Quadratic Formula, which will likely be easier than completing the square. Factoring won't work because you know the answer will have a radical in it.

The Quadratic Formula is x = (-b + √(b² - 4ac) ) / (2a), where a, b and c are the coefficients of the terms.

x = (-(-10) + √((-10)² - 4(3)(5)) ) / (2(3))
x = (10 + √(100 - 60) ) / (6)
x = (10 + √(40) ) / (6)
x = (10 + 2 √(10) ) / (6)
x = 5/3 + 1/3 √(10)

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

3y + 2x < 15
y - x > 1

State the coordinates of a point in the solution to this system. Justify your answer.

Answer:

Rewrite the inequalities into slope-intercept form and put them into your calculator. Check the table of values for points on the line.

The first inequality will have a solid line. The second will have a broken or dashed line. Points on the broken line are NOT in the solution set.

You're going to graph below the first line and above the second line.

3y + 2x < 15
3y < -2x + 15
y < -2/3x + 5

y - x > 1
y > x + 1

You're graph should look like this:

Pick any point in the double-shaded area that is NOT on the broken line.

For example, (-10,0) is in the solution set because it's in the double-shaded area where the big S is labeled.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

June 2024 Algebra Regents Part II

This exam was adminstered in January 2024 .

June 2024 Algebra, Part II

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

25. Solve 5(x - 2) < 3x + 20 algebraically.

Answer:

Distribute the 5 and use Inverse Operations.

5(x - 2) < 3x + 20
5x - 10 < 3x + 20
5x < 3x + 30
2x < 30
x < 15

This wasn't a "trick" question which required flipping the direction of the inequality symbol because no negatives where multiplied or divided.

26. Given g(x) = x₃ + 2x² - x, evaluate g(-3).

Answer:

Substitute and evaluate. Remember the order of operations. Or put it into your calculator.

g(-3) = (-3)₃ + 2(-3)² - (-3) = -6

If you put this into your calculator, remember to use all the parentheses that I used above to avoid a mistake.

You could also graph y = x₃ + 2x² - x and check the table of values for x = -3.

Whatever you do, write something on the paper other than just the answer, which is only worth one point. I suggest you write down the substition. It will be assumed that you calculated it in your head or with a calculator, so you don't need to write down every step.

27. Given the relation R = {(-1,1), (0,3), (-2,-4), (x,5)}.
State a value for x that will make this relation a function.
Explain why your answer makes this a function.

Answer:

The relation is a function as long as the x values do not repeat.

This means you can write any number except -1, 0, or -2.

That is, you can write x = 2, x = 10, x = π, x = 1/2 or pretty much anything you want, so long as it's a number other than the three already in the set.

28. A survey of 150 students was taken. It was determined that 2/3 of the students play video games.
Of the students that play video games, 85 also use social media.
Of the students that do not play video games, 20% do not use social media.

Complete the two-way frequency table

Answer:

Start with the numbers you know and work backward to find the missing information.

The grand total, bottom right corner, is 150. If 2/3 play video games, that means that the bottom row will be divided into 100 and 50.

Of the students who play video games (100 students), 85 use social media, which means that 100 - 85 = 15 do not.

Of the ones not playing video games (50 students), 20% don't use social media. Since .20 * 50 = 10, that means 50 - 10 = 40 do use social media.

Add the rows across. You will get the following table:

29. Use the method of completing the square to determine the exact values of x for the equation x² + 10x - 30 = 0.

Answer:
They stated that you must complete the square. If you use any other method, then the most you can get is 1 point for a correct value of x.

To complete the square, you need to divide the middle value by 2, which is 10/2 = 5. That means that (x + 5)² will be part of the solution. To complete the square you need to add 25 to each side of the equation.

x² + 10x - 30 = 0
x² + 10x = 30
x² + 10x + 25 = 55
(x + 5)² = 55
x + 5 = +√(55)
x = -5 +√(55)

30. Factor 20x³ - 45x completely.

Answer:
Whenever you see "Factor completely", you can be pretty sure that there will be more than one step. Show them all.

First, look for common factors in each of the terms. Then factor what remains. There are two terms separated by a minus sign, so keep an eye out for a Difference of Squares.

20x³ - 45x = 5x (4x² - 9) = 5x (2x + 3)(2x - 3)

Note that this is an expression and not an equation. Do NOT attempt to "solve" it, or you will lose a point.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Geometry Problems of the Day (Geometry Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Geometry Regents

Part I

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


17. If ABCD is a parallelogram, which additional information is sufficient to prove that ABCD is a rectangle?

(1) AB ≅ BC
(2) AB || CD
(3) AC ≅ BD
(4) AC ⊥ BD

Answer: (3) AC ≅ BD

A rectangle is a parallelogram with right angles. One additional property of a rectangle is that its diagonals are congruent, but they are not necessary perpendicular.

Choice (1) makes the parallelogram a rhombus, not a rectangle. Eliminate Choice (1).

Choice (2) is true of every parallelogram. Eliminate Choice (2).

Choice (3) is true of rectangles. This is the correct answer.

Choice (4) is there to trick you. The sides being perpendicular, AB ⊥ BC, etc, are true of rectangles. However, the diagonals are only perpendicular if it is a rhombus (including squares). Eliminate Choice (4).

18. Line segment APB has endpoints A(-5,4) and B(7,-4). What are the coordinates of P if AP:PB is in the ratio 1:3?

(1) (-2,2)
(2) (-1,1.3)
(3) (1,0)
(4) (4,-2)

Answer: (1) (-2,2)

A ratio of 1:3 means that the line segment can be broken up into four equal (congruent) segments. AP has the legnth of one of those segments and PB has the length of three of those segments.

Because they used the ratio 1:3, point P is 1/4 of the way from A to B. That means, for this one problem, you can use a shortcut. (The "real" way is below.) You can find the midpoint of AB, call it M. Then find the midpoint of AM, which will be point P because 1/2 of 1/2 is 1/4.

M = ( (-5+7)/2, (4+(-4))/2 ) = (1,0)

P = ( (-5+1)/2, (4+0/2) ) = (-2,2), which is Choice (1).

The "right" way, and the way you would have to do it if the numbers weren't so nice is as follows:

x: 7 - (-5) = 12, and 1/4 (12) = +3.

y: -4 - 4 = -8, and 1/4 (-8) = -2

P = (-5 + 3, 4 - 2) = (-2, 2), which is Choice (1).

19. In the diagram below, AB and CD intersect at E, and CA and DB are drawn.

If CA || BD, which statement is always true?

(1) AE ≅ BE
(2) AE ≅ BE
(3) △AEC ~ △BED
(4) △AEC ≅ △BED

Answer: (3) △AEC ~ △BED

While the triangles are drawn to look like they are congruent, there is not enough information to same that they are congruent.

However, there are vertical angles (which are congruent) at point E, and because CA || BD, there are alternate interior angles that are congruent. That makes the triangles similar by AA Theorem. Choice (3) is correct.

Note that if the triangles were congruent, then they would also be similar, so two choices would be correct because it doesn't say "similar but not congruent".

Also, note that if Choice (1) was true, then Choice (3) would be true by ASA, which would make Choice (2) true as well by CPCTC.

You know that all four choices cannot be true.

20. If sin(3x + 9)° = cos(5x - 7)°, what is the value of x?

(1) 8
(2) 11
(3) 33
(4) 42

Answer: (2) 11

The following is always true: sin (x) = cos (90 - x).

Therefore,

3x + 9 = 90 - (5x - 7)
3x + 9 + 5x - 7 = 90
8x + 2 = 90
8x = 88
x = 11

Checking out work: 3(11) + 9 = 42, 5(11) - 7 = 48, and 42 + 48 = 90

Choice (2) is the correct answer.

21. Which set of integers could represent the lengths of the sides of an isosceles triangle?

(1) {1,1,3}
(2) {2,2,5}
(3) {3,3,6}
(4) {4,4,7}

Answer: (4) {4,4,7}

For a triangle to be isosceles, two of the sides must be the same length. This is true of all four choices.

However, for three lines to make a triangle, the sum of the lengths of the two shorter sides must be longer than the length of the longest side. Longer, not equal.

Choices (1), (2), and (3) do not make triangles. The sides are too short. Only Choice (4) makes a triangle, so it is the correct answer.

22. In the diagram shown below, altitude CD is drawn to the hypotenuse of right triangle ABC.

Which equation can always be used to find the length of AC?

(1) AC/CD = CD/AD
(2) CD/AC = AC/AB
(3) AC/CD = CD/BC
(4) AB/AC = AC/AD

Answer: (4) AB/AC = AC/AD

An altitude drawn from a right angle to the hypotenuse of a right triangle will divide the right triangle is such a way that the two smaller triangles will be similar to each other and to the original triangle. This is the Right Triangle Altitude Theorem.

This means that the corresponding sides of the triangles are proportional.

Choice (1) only uses sides from one triangle. It doesn't use corresponding pairs of sides. Eliminate Choice (1).

In Choice (2), CD doesn't correspond to AC in triangle ABC, so the proportion is no good. Eliminate Choice (2).

In Choice (3), the proportion says hypotenuse / short leg = long leg / hypotenuse. Elimnate Choice (3).

Choice (4) has hypotenuse / long leg = hypotenuse / long leg. This is the correct answer.

23. Which congruence statement is sufficient to prove parallelogram MARK is a rhombus?

(1) MA ≅ MK
(2) MA ≅ KR
(3) ∠K ≅ ∠A
(4) ∠R ≅ ∠A

Answer: (1) MA ≅ MK

To show that a parallelogram is also a rhombus, you need to show that either the consecutive sides are congruent or that they diagonals are perpendicular. (The diagonals do not have to be congruent, and will only be congruent in a square or rectangle.)

There are no choices suggesting that the diagonals are perpendicular, so that means congruent sides.

Choice (1) has consecutive sides MA and MK. This is the correct answer.

Choices (2) and (3) are true for any parallelogram.

If Choice (4) is true in a parallelogram, then the parallegram is a rectangle because the angles would both have to be 90 degrees because the sum of angles A and R is 180.

24. A line whose equation is y = -2x + 3 is dilated by a scale factor of 4 centered at (0,3). Which equation represents the image of the line after the dilation?

(1) y = -2x + 3
(2) y = -2x + 12
(3) y = -8x + 3
(4) y = -8x + 12

Answer: (1) y = -2x + 3

Dilating a line will NOT change its direction which means that the slope DOES NOT change. Eliminate Choices (3) and (4) immediately.

When you dilate a line, you either get a line that is parallel to the original line OR you get the original line if the center of dilation was a point on that line.

Since (0,3) is a point on the line y = -2x + 3 (Proof: 3 = -2(0) + 3), the image will be the same line as the pre-image. So Choice (1) is the correct answer.

End of Part I.

How did you do?

Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Geometry Problems of the Day (Geometry Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Geometry Regents

Part I

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


9. In circle O below, chords CT and EN intersect at point A. Chords CB and NT are drawn.

Which statement is always true?

(1) NT/TA = CB/BA
(2) ∠BAC ≅ ∠ATN
(3) NA/AB = TA/AC
(4) ∠BCA ≅ ∠NTA

Answer: (1) NT/TA = CB/BA

Once again the triangles are similar (not congruent). If they weren't congruent, then there would be no answer to this question. This means that the corresponding angles are congruent and the corresponding sides are proportional.

You can see that the triangles are similar because there are vertical angles at point A which must be congruent, and angle C and angle N are both inscribed angles that intercept the same arc, BT, so they must be congruent. By AA, the triangles are similar.

Choice (1) says NT/TA = CB/BA. These are two pairs of corresponding sides in the correct order. This statement is true. Choice (1) is the correct answer.

In Choice (2), a vertical angle is being compared to an inscribed angle. There is no reason for these to be congruent. Eliminate Choice (2).

In Choice (3), the sides are not listed correctly. In fact, if there was multiplication instead of division, the statement would've been correct. Eliminate Choice (3).

Choice (4) compares angles C with angle T. As we have alreday said, angle C is congruent to angle N, not T. Angle T is congruent to angle B because both are inscribed angles that mark off arc CN. Eliminate Choice (4).

10. In the diagram below of △ABC, CBF is drawn, AB bisects ∠FBD, and BD ⟂ AC.

If m∠C = 42^o, what is m∠A?

(1) 24^o
(2) 33^o
(3) 48^o
(4) 66^o

Answer: (1) 24^o

Follow along on the diagram below. You have to hit most of the angles in the diagram to get the answer.

Angle C is 42 degrees. Angle CDB is 90 degrees because BD is perpendicular to AC. That means that angle CBD is 48 degrees because CBD is a triangle and the angles must total 180 degrees.

Angles CBD, DBA, and ABF form a straight line and must add up to 180 degrees. Because AB bisects FBD, then angles DBA and ABF must be equal in measure, so we can label them both x.

That means 42 + x + x = 180
42 + 2x = 180
2x = 132
x = 66

If angle DBA = 66 and angle BDA = 90, then angle A = 180 - 90 - 66 = 24 degrees, which is Choice (1).

11. In circle O below, OA = 6, and m∠COA = 100°.
What is the area of the shaded sector?

(1) 10 π
(2) 26 π
(3) 10 π / 3
(4) 26 π / 3

Answer: (2) 26 π

The area of the circle is A = π r². The area of the shaded sector is the area of the circle times the fraction of the circle that's shaded in.

The central angle is 100°, and that sector is NOT shaded in. That means that the shaded portion is 260°/360° or 260/360, which simplifies to 13/18. (If you're using a calculator, you don't need to simplify).

Leave the answer in terms of pi (That is, don't hit the pi key on the calculator or enter a number for it) because all the answers have pi in them.

A = 260/360 (6)² = 26

The correct answer is Choice (2).

12. In rectangle ABCD, diagonal AC is drawn. The measure of ∠ACD is 37° and the length of BC is 7.6 cm. What is the length of AC, to the nearest tenth of a centimeter?

(1) 4.6
(2) 6.1
(3) 10.1
(4) 12.3

Answer: (4) 12.3

If you sketch a little diagram, you will see that 7.6 is opposite the 37 degree angle and that the diagonal is the hypotenuse of a right triangle.

Opposite and hypotenuse means that you need to use the sine ratio.

sin 37 = 7.6 / x
x = 7.6 / sin 37 = 12.62...

The correct answer is Choice (4).

Note that if you use cosine or tangent, you will likely get one of the other choices.

Also, since the hypotenuse MUST BE longer than 7.6, you could immediately eliminate Choices (1) and (2). They are too short.

13. A peanut butter manufacturer would like to use a cylindrical jar with a volume of 1180 cm³. The jar has a height of 10 cm. What is the diameter of the jar, to the nearest tenth of a centimeter?

(1) 3.8
(2) 6.1
(3) 10.9
(4) 12.3

Answer: (4) 12.3

Notice that they asked for the diameter and not the radius. The formula uses radius, and that is you must solve for first. Then you can get the diameter.

The formula for volume of a cylinder is V = π r² h. Plug in what you know to find out what you don't know.

1180 = (3.141592)(r)²(10)
r² = 1180 / ((3.141592)(10))
r = √(1180 / ((3.141592)(10)) = 6.128...

So the diameter is 6.128 * 2 = 12.257... or 12.3, which is Choice (4).

14. Triangle KLM is dilated by a scale factor of 3 to map onto triangle DRS. Which statement is not always true?

(1) ∠K ≅ ∠D
(2) KM = 1/3 DS
(3) The area of triangle DRS is 3 times the area of triangle KLM.
(4) The perimeter of triangle DRS is 3 times the area of triangle KLM.

Answer: (3) The area of triangle DRS is 3 times the area of triangle KLM.

When you dilatte a polygon, length (such as perimeter) will be increased by the scale factor (multiplied by the scale factor). However, the area will be the original area times the square of the scale factor, because both the length and the width have increased.

The correct choice is Choice (3).

The corresponding angles will remain congruent because the two triangles are similar. And the legnth of side DS will be 3 times bigger than side KM, so KM is 1/3 DS.

15. A rectangle with dimensions of 4 feet by 7 feet is continuously rotated about one of its 4-foot sides. The resulting three-dimensional object is a

(1) cylinder with a height of 7 feet and a base radius of 4 feet.
(2) cylinder with a height of 4 feet and a base radius of 7 feet.
(3) cone with a height of 7 feet and a base radius of 7 feet.
(4) cone with a height of 4 feet and a base radius of 7 feet.

Answer: (2) cylinder with a height of 4 feet and a base radius of 7 feet.

If a rectangle is rotated about an axis, there is no way that they resulting shape could be a cone. It will be a cylinder. Eliminate Choices (3) and (4).

If it is rotated about the 4-foot side then 4 feet will be the height of the cylinder. The rectangle will swing in a circle with a radius of 7 feet.

The correct answer is Choice (2).

16. In right triangle ABC, altitude CD is drawn to hypotenuse AB. If AD = 4 and CD = 8, the length of BD is

(1) √(48)
(2) √(80)
(3) 12
(4) 16

Answer: (4) 16

The Right Triangle Altitude Theorem tells us that (CD)² = (AD)(BD). Substitute and solve.

(8)² = (4)(x)
64 = 4x
x = 16

The correct answer is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Geometry Problems of the Day (Geometry Regents, June 2024 Part I)

This exam was adminstered in June 2024.

More Regents problems.

June 2024 Geometry Regents

Part I

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


1. In the diagram below, △BRI is the image of △JOE after a translation. Triangle CAT is the image of △BRI after a line reflection.

Which statement is always true?

(1) ∠R ≅ ∠T
(2) ∠J ≅ ∠A
(3) JE ≅ RI
(4) OE ≅ AT

Answer: (4) OE ≅ AT

You can follow the transformations, and if you wish to, you can mark up the book to show which points are moving to which new poisitions. Then you can see which angles are corresponding (and therefore congruent) and which sides are corresponding (and congruent).

But here's the little secret for state exams. Keep in mind that this doesn't always work in your classroom because teachers are sloppy (and I can be guilty of this myself sometimes if I'm not paying attention or if I'm just tired or overworked). The order of the letter is the triangle name is important. They will always correspond on state exams, and they should in textbook examples. (Textbook writers aren't always 100% at the top of their game.).

So B goes to J goes to C, R goes to O goes to A, and I goes to E goes to T.

You can eliminate Choices (1) and (2) immediately. Careful checking shows that Choice (3) is wrong and Choice (4) is the correct answer because JOE goes to CAT.

2. right cylinder is cut parallel to its base. The shape of this cross section is a

(1) cone
(2) circle
(3) triangle
(4) rectangle

Answer: (2) circle

If it is cut parallel to the base, then the cross-section has to be a circle, which is the same shape as the base of the cylinder. The correct answer is Choice (2).

A rectangle could occur if the cross section was perpendicular to the base.

A triangle would not happen, no matter how you slice it. And a cross section is a 2-D shape, so it could not be a cone, which is 3-D.

3. What is the minimum number of degrees that a regular hexagon must rotate about its center to carry it onto itself?

(1) 45°
(2) 72°
(3) 60°
(4) 120°

Answer: (3) 60°

Any figure will carry onto itself if you rotate it 360 degrees. Because regular polygons have symmetry, you only have to rotate them a minimum of 360 divided by the number of sides that the polygon has, which in this case is 6.

360 / 6 = 60 degrees, so the correct answer is Choise (3).

120 degrees is the size of each internal angle, but that's not the question that's being asked.

4. In the diagram below, a sphere is inscribed inside a cube. The cube has edge lengths of 18.

What is the volume of the sphere, in terms of π?

(1) 108π
(2) 432π
(3) 972π
(4) 7776π

Answer: (3) 972π

Use the formula for the volume of a sphere. The length of the box is the diameter of the ball. You need to cut that in half to get the radius.

Note that one of the wrong choices will always be the result if a student uses the diameter instead of the radius. They expect some students to do this. As a result, you can be fairly certain that the largest choise will not be the correct answer.

V = 4/3 π r³ = 4/3 π (9)³ = 972 π, which is Choice (3).

5. In the diagram below, EM intersects HA at J, EA ⟂ HA, and EM ⟂ HM.
If EA = 7, EJ = 9, AJ = 5.4, and HM = 3.29, what is the length of MJ, to the nearest hundredth?

(1) 2.47
(2) 2.63
(3) 4.11
(4) 4.39

Answer:(1) 2.47

You can prove that the two traingles are similar in shape and therefore their corresponding sides are proportional. You don't have to prove anything though because if this wasn't true, there wouldn't be any way to solve this question. So in this particular problem if you just assumed that they were similar, you would be okay. Because they are.

Briefly, each triangle has a right angle, which are congruent to each other, and the vertical angles are congruent. Therefore, by AA, the triangles must be similar. (They are not congruent.)

Write down the lengths that you know and put an x next to the side you're looking for, MJ.

Comparing sides, you will see that

5.4 / x = 7.2 / 3.29

so 7.2 x = (5.4)(3.29)

and x = (5.4)(3.29)/7.2 = 2.4675

To the nearest hundredth, MJ = 2.47, which is Choice (1).

6. Which equation represents the line that passes through the point (2,-7) and is perpendicular to the line whose equation is y = 3/4x + 4?

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

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

Parallel lines have the same slope. Perpendicular lines have slopes that are NOT the same but are inverse reciprocals -- the product of the slopes is -1.

Since they want perpendicular, you can cross out Choices (1) and (2).

The question uses slope-intercept form, which gives you the slopes and the y-intercept. The choices are given using point-slope form, which gives you the slope and a point on the line.

The thing to remember is that the formula has minus signs in it (as do many, many formulas in Algebra and Geometry): y - k = m(x - h).

You may have learned that formula as y - y₀ = m(x - x₀), which is the same thing. However, if you get used to seeing h and k, you'll notice that (h,k) is all over the place.

Since the point is (2,-7) and the signs are "flipped" in the equation, then the correct answer is Choice (3).

7. In △RHM below, m∠R = 110° and m∠M = 40°.

If △RHM os reflected over side HM to form quadrilateral RHR'M, which statement is always true?

(1) Quadrilateral RHR'M is a parallelogram.
(2) m∠MHR' = 40°
(3) m∠HMR' = 40°
(4) MR ≅ HR'

Answer: (3) m∠HMR' = 40°

You can make a sketch in the book underneath the original image and mark it up.

You can see that the correct answer is Choice (3) because HMR and HMR' are corresponding angles of congruent triangles.

The figure is not a parallelogram because the opposite sides are not parallel. It's a kite. Eliminate Choice (1).

The measure of angle MHR' is the same as angle MHR, both of which can be calculated to be 30 degrees. (180 - 110 - 40 = 30). Choice (2) is incorrect.

Since the figure is a kite and not a parallelogram, the opposite sides are not congruent, so MR is not congruent to HR'. This eliminates Choice (4).

8. The funnel shown below can be used to decorate cookies with melted chocolate. The funnel can be modeled by a cone whose radius is 6 cm and height is 13 cm.
The baker uses 2 cubic centimeters of chocolate to decorate each cookie. When the funnel is completely filled, what is the maximum number of cookies that can be decorated with the melted chocolate?

(1) 78
(2) 245
(3) 490
(4) 735

Answer: (2) 245

Find the volume of the funnel using the formula for the volume of a cone: V = 1/3 π r² h.

V = (1/3) π (6)² (13) = 490.08845396...

If the baker uses 2 cm³ per cookie, then divide the result by 2: 490 / 2 = 245.

The correct answer is Choice (2).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories. Order the softcover or ebook at Amazon. And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!
Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.