Allergies

(Click on the comic if you can't see the full image.)

(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Based on a true story.

I've actually worked on a few comic for use in my classroom over the past few months, and I'll rework them for use on this page. I've been teaching a Graphic Novel course that is an Art credit even though I'm neither Art nor English teacher. We're learning about them and their history, and trying to create some of our own, without worrying to much about the actual artwork. It's more about the effort and design we're putting into them. For anyone really interested, developing art skills (some already have them) will come later. Or they can tell their story by partnering with an artist, and they'll be able to explain exactly what they envision.

Much of what follows appears on the page from the last comic I published -- except that I published it quite a while after the date on the comic.

The slightly longer story: every time I looked at the blog after that, the old "imposter syndrome" reared its ugly head. And then there's the joke asking if you could still be paranoid if people really are out to get you, and I wondered if that applied to with this syndrome as well. It's not like I ever achieved major success with this blog, and for the time that I did, I really didn't know how to capitalize on that to take it to another level (like, for example, Math with Bad Drawings).

And now blogs are past their heyday. Even now, my Regents pages, which always got more hits than my comics, are down in views, as search engine AI spit out answers before they even point to my pages. Fewer of today's high schoolers would even know I exist.

So that's what's been going on. I've been in a decline pretty much since I knew I was closing in on my 2,000th strip and I would need to do something for it. I think I'm past the point of doing something BIG, not because no one is expecting much, but because few seem to be out there at all. And I did MS Paint myself into a corner with a breakup story that was supposed to be resolved fairly quickly -- but I had to write it and then edit down what I was thinking, and then draw it. I hope I will get to that because I don't want to leave those characters hanging. (They could eventually reconcile off-screen.)

If you are out there, I will remind you of the many times that I've said, "I thrive on feedback." And then I might point out that I get so little of it.

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

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

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


17. Consider the system of equations below.

3x + 2y = 1
2y + z = 2
2x - 2z = -6 Given this information, what is P(F and I), the probability that a randomly selected teenager uses both websites?

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

Answer: (1) 1

Subtract the second equation from the first equation and the y variable will disappear. Now you can solve for x.

3x + 2y = 1
2y + z = 2
2x - 2z = -6

3x - z = -1
2x - 2z = -6

6x - 2z = -2
2x - 2z = -6

4x = 4

x = 1

This is Choice (1).

Checking: 2(1) - 2z = -6; -2z = -8, z = 4

Then, 2y + 4 = 2, 2y = -2, y = -1

Finally, 3(1) + 2(-1) = 1. Check!

18. The point (2,–3) lies on the graph of the equation y = f(x). Which point must lie on the graph of the equation y = f(x - 4) + 1?

(1) (1,1)
(2) (-2,-2)
(3) (3, 7)
(4) (6,-2)

Answer: (4) (6,-2)

The -4 inside the parentheses shifts the graph four units to the right. The +1 outside the parentheses shifts the graph on unit up.

If you translate the point (2,-3) four units to the right and one unit up, you end up at (6,-2).

The correct answer is Choice (4).

19. Which statement best describes the end behavior of the function y = log(x - 3)?

(1) As x → -∞, y → -∞, and as x → ∞, y → ∞.
(2) As x → 3, y → -∞, and as x → ∞, y → ∞.
(3) As x → -∞, y → 0, and as x → ∞, y → ∞.
(4) As x → 3, y → 0, and as x → ∞, y → ∞.

Answer: (2) As x → 3, y → -∞, and as x → ∞, y → ∞.

Graph the function. One the left side, the graph goes to negative infinity as it gets closer to 3. On the right side, both x and y tend toward infinity.

Choice (2) is the correct answer.

20. The black bear population for a certain area of the Adirondacks can be modeled by the equation

B = 5835.943(1.026)^t, where t is measured in years since 2010. Kieran would like to rewrite this model in terms of a 5-year growth rate. Kieran’s model is best represented by

(1) B = 5835.943(1.005147)^t/5
(2) B = 5835.943(1.005147)^5t
(3) B = 5835.943(1.136938)^t/5
(4) B = 5835.943(1.136938)^5t

Answer: (3) B = 5835.943(1.136938)^t/5

If there is a 5-year growth rate, then the exponent will be divided by 5, or t/5. Eliminate Choices (2) and (4).

(1.026)^t is equal to (1.026⁵)^t is equal to (1.136938)^t/5.

This is Choice (3).

21. Which expression or expressions are equal to 0 for all real numbers?

I. (x² + y²)² + (x² + y²)² - 2(x² + y²)² II.(x² + y²)² - (x² + y²)² III. (x² + y²)² - (x² + y²)² - (2xy)²

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

Answer: (4) I and III, only

The first expression is "obviously" equal to 0 (see below), but you need to expand the second and third choices to see if all the terms will cancel out. Again, it is "obvious" that they both can't be true. However, it is possible that neither one is true.

In the first expression, let z = (x² + y²)². So z + z - 2z = 0, which is true. Eliminate Choice (2).

In the second expression, the first term is not the same as the second term, so subtracting them cannot equal 0. Eliminate Choice (3).

Expand the third expressiong.

(x² + y²)² - (x² + y²)² - (2xy)²
(x⁴ + 2x²y² + y⁴) - (x⁴ - 2x²y² + y⁴) - (4x²y²)
x⁴ + 2x²y² + y⁴ - x⁴ + 2x²y² - y⁴ - 4x²y²
x⁴ - x⁴ + 2x²y² + 2x²y² + y⁴ - 4x²y² - y⁴ = 0

Choice (4) is the correct answer.


22. The equation 1/x - 1/5 = x/5 has

(1) rational solutions
(2) irrational solutions
(3) imaginary solutions
(4) no solutions

Answer: (2) irrational solutions

Solve for x.

1/x - 1/5 = x/5

1/x = x/5 + 1/5

1/x = (x + 1)/5

x(x + 1) = 5

x² + x = 5

x² + x - 5 = 0

If you check the discriminant, b² - 4ac, then 1² - (4)(1)(-5) = 21, which is positive but not a perfect square. This means that there are two solutions but they will be irrational.

This is Choice (2).

23. For x ≠ +4y, the expression (x² + 3xy - 28y²) / (16y² - x² is equivalent to
(1) -1 - (7/4) y
(2) (x - 7y) / (4y - x)
(3) (x + 7y) / (x + 4y)
(4) (-x - 7y) / (x + 4y)

Answer: (4) (-x - 7y) / (x + 4y)

Factor the numerator and the denominator and simplify. (x² + 3xy - 28y²) / (16y² - x²
( (x + 7y)(x - 4y) ) / ( (4y - x)(4y + x) )
( (x + 7y)(-1) ) / (4y + x)
( -x - 7y) / (x + 4y)

This is Choice (4).

24. Which equation represents a parabola with a focus of (-2,1) and directrix of y = 5?,

(1) (x + 2)² = -8(y - 3)
(2) (x + 2)² = 5(y - 1)
(3) (x + 2)² = -8(y - 1)
(4) (x + 2)² = 8(y - 3)

Answer: (1) (x + 2)² = -8(y - 3)

The formula for finding a parabola from the focus and directrix is (x - h)² = 4p(y - k), where p is the distance from the vertex to the focus. The vertex is halfway between the focus and directrix, or (-2, 3), which makes p = -2.

Substituting what we know, we get (x - (-2))² = 4(-2)(y - 3), or (x + 2)² = -8(y - 3).

This is Choice (1).

End of Part I.

Questions, comments, and corrections welcome.

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

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

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

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


9. The probabilities that a randomly selected teenager uses social media websites F and I are shown below.

P(F) = 0.71
P(I) = 0.52
P(F or I) = 0.77 Given this information, what is P(F and I), the probability that a randomly selected teenager uses both websites?

(1) 0.06
(2) 0.19
(3) 0.46
(4) 0.96

Answer: (3) 0.46

The union (OR) of two probabilities is the equal to the sum of the independent probabilities minus their intersection (AND).

P(F or I) = P(F) + P(I) - P(F and I)
P(F) + P(I) - P(F or I) = P(F and I)
0.71 + 0.52 - 0.77 = P(F and I)
0.46 = P(F and I)

This is Choice (3).

10. Consider f(x) = (x - 2)²(x + 3), and g(x) as strictly defined in the table below.
Which statement or statements must be true, based on the information given?
I. Both f(x) and g(x) have the same x-intercepts.
II. Both f(x) and g(x) have a y-intercept at y = -6

(1) I, only
(2) II, only
(3) I and II
(4) neither I or II

Answer: (1) I, only

The x-intercepts of f(x) are 2 and -3. The x-intercepts of g(x) are -3 and 2. These are the same.

The y-intercept of f(x) = (-2)²(3) = 12. The y-intercept of g(x) is -6. These are not the same.

The correct answer is Choice (1).

11. Josie examines the graphs of f(x) = 3^x - 8 and g(x) = 1 / (x² - 4). The number of solutions to f(x) = g(x) is

(1) 1
(2) 2
(3) 3
(4) 0

Answer: (3) 3

Just graph them. The answer is 3.

Choice (3) is the correct answer.

12. Which binomial is a factor of g³ + 6g² + g - 14?

(1) g - 1
(2) g - 2
(3) g + 1
(4) g + 2

Answer: (4) g + 2

Once again: graph it. Check the zeroes. The roots are x = -2, or x = -2 + √(11).

If g - 1 is a factor, then the polynomial will eqaul zero when g = 1. However, it is -6. Eliminate Choice (1).

If g - 2 is a factor, then the polynomial will eqaul zero when g = 2. However, it is 20. Eliminate Choice (2).

If g + 1 is a factor, then the polynomial will eqaul zero when g = -1. However, it is -10. Eliminate Choice (3).

If g + 2 is a factor, then the polynomial will eqaul zero when g = -2. And it is 0. Choice (4) is the correct answer.

13. Consider the recursively defined sequence below.

a₁ = 8
a_n = 2a_n-1 Which explicit formula represents the same sequence?

(1) a_n = 2ⁿ
(2) a_n = 2(4ⁿ)
(3) a_n = 2⁽ⁿ⁺²⁾
(4) a_n = 8ⁿ

Answer: (3) a_n = 2⁽ⁿ⁺²⁾

If the answer doesn't jump out at you, write out the first few terms of the sequence. Then check each one.

The recursive formula is doubling, starting with 8: 8, 16, 32, 64, ...

Choice (1) is 2, 4, 8, ... Eliminate Choice (1).

Choice (2) is 8, 32, 128, ... Eliminate Choice (2).

Choice (3) is 8, 16, 32, 64. ... Choice (3) is the correct answer.

Choice (4) is 8, 64 ... Eliminate Choice (4).


14. What is the exact value of tan(-5π/6)?

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

Answer: (1) 1 / √(3)

Tangent is positive in Quadrants I and III and this is Quadrant III. Eliminate Choices (2) and (4).

The coordinates for that point on the unit circle are (-√(3)/2, -1/2).

Tangent is sin/cos = (-1/2) / (-√(3)/2) = 1 / √(3)

This is Choice (1).

15. Given m ≠ 0 and (17^1/m)ⁿ = 17², what is n in terms of m?
(1) 2m
(2) 2/m
(3) m/2
(4) 2^m

Answer: (1) 2m

Multiplying the exponents give you 17^n/m = 17². This means n/m = 2.

Therefore, n = 2m.

This is Choice (1).

16. In order to qualify for a college tennis scholarship, Joe needs to win 90% of the matches he plays during his senior year of high school. If he has won 8 of the 10 matches that he has played, which equation can be used to determine how many more consecutive matches, x, Joe must win in order for his winning percentage to equal 90%?

(1) (8 + x) / x = 0.90
(2) 8 / (10 + x) = 0.90
(3) 8/10 + x = 0.90
(4) (8 + x) / (10 + x) = 0.90

Answer: (4) (8 + x) / (10 + x) = 0.90

The number of games won divided by the number of games played must equal 0.90.

Currently, Joe is at 8/10 = .80. If he keeps winning, his average will be 9/11, 10/12, 11/13, etc.

Both the numerator and the denominator will increase.

This means that x games will be added to both the 8 and the 10, so Choice (4) 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 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, June 2025 Part I)

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

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


1. Which expression is equivalent to 2c³√(c) ?

(1) 2c^4/3
(2) 2c^3/4
(3) (2c)^4/3
(4) (2c)^3/4

Answer: (1) 2c^4/3

When there's a fraction as an exponent, the numerator is the power the base is raised to while the denominator is the nth root of the base.

2c³√(c) = 2c (c)^1/3

When you multiply variables, keep the base and add the exponents. If you add 1 and 1/3, the sum is 4/3. Therefore,

2c³√(c) = 2c (c)^1/3 = 2c^4/3

The correct answer is Choice (1).

2. Which investigation technique is most often used to determine the cause and effect of a medication?

(1) observational study
(2) survey
(3) controlled experiment
(4) census

Answer: (3) controlled experiment

To determine cause and effect, you need to conduct a controlled experiement. A survey, census or observational study are insufficient.

The correct answer is Choice (3).

3. What is the solution to 5(2)^19x = 50?

(1) x = (log(50)) / 19
(2) x = (log₂(10)) / 19
(3) x = (log₂(45)) / 19
(4) x = 5 / 19

Answer: (2) x = (log₂(10)) / 19

Use inverse operations.

5(2)^19x = 50
(2)^19x = 10
log₂ 10 = 19x
x = log₂ 10 / 19

The correct answer is Choice (2).

4. The function P(t) = 256,485(0.965)^t models the decreasing population of a city from 1999 to 2014, where t is the time in years since 1999.

Which statement is not true?

(1) The function estimated the population was 256,485 in 1999.
(2) The decay rate was 0.35%.
(3) The decay factor is 0.965.
(4) The population declined over 15 years.

Answer: (2) The decay rate was 0.35%.

I will say that this was a confusing question, and for a moment, I wondered if there was a typo. However, I would've remembered if a recent question had been invalidated. My trouble comes from them using the term "decay factor", which I have to say, personally, I don't see that term used. Note that I don't normally teach Algebra II, but when talking about equations of this type, we usually talk about the "decay rate" and not the "decay factor".

In any case ...

In Choice (1), the starting population is 256,485 in 1999. This is true, so eliminate Choice (1).

In Choice (2), the decay rate is NOT 0.35%. If you subtract 1.000 - 0.965, you get 0.035, which is 3.5%. This is the correct answer.

In Choice (3), the decay factor is 0.965. This is true. This is different from the decay rate. This is the factor that you are multiply the starting value by. Eliminate Choice (3).

In Choice (4), since the decay factor is less than 1.00, which is why it's decay and not growth, the population will continually shrink. The question says it's decreasing from 1999 to 2014, so it's decreasing for 15 years. This is correct, so eliminate Choice (4).

5. Four different surveys gathered data about the purchasing behaviors of pet owners. Pet owners from the same population were randomly selected. While collecting data, Chris surveyed 942 pet owners, John surveyed 410, Brooke surveyed 800, and Shane surveyed 100. Whose survey will likely have the smallest margin of error?

(1) Brooke
(2) Chris
(3) John
(4) Shane

Answer: (2) Chris

The more pet owners interviewed, the smaller the margin of error that can be expected. More is better.

Chris interviewed the most. This is Choice (2).

6. Given i is the imaginary unit and a = i³, b = i², and c = i, which expression is equivalent to 2ax² + 3bx - cx?

(1) -2ix² - 3x + ix
(2) -2ix² - 3x
(3) -2ix² - 3x - ix
(4) -8ix² - 3x - ix

Answer: (3) -2ix² - 3x - ix

Replace a with (-i). Replace b with (-1). Replace c with (i). Simplify the expression.

2ax² + 3bx - cx = 2(-i)x² + 3(-1)x - (i)x = -2ix² - 3x - ix.

This is Choice (3).

7. Which sequence has a common ratio of 1/2?

(1) -1/4 a, -1/8 a, -1/16 a, -1/32 a ...
(2) 1/32 a, 1/16 a, 1/8 a, 1/4 a, ...
(3) 20a, 39/2 a, 19a, 37/2 a, ...
(4) 22a, 22.5a, 23a, 23.5a, ...

Answer: (1) -1/4 a, -1/8 a, -1/16 a, -1/32 a ...

Multiply the first term by 1/2. Did you get the second term? If not, eliminate that choice. If you did, check the next number.

In Choice (1), if you multiply (-1/4a)(1/2) = -1/8 a, etc. This is the Correct answer.

In Choice (2), if you multiply (1/32 a)(1/2) = 1/64 a. The common ratio in the second choice is 2, not 1/2. Eliminate Choice (2).

In Choice (3), half of 20a is 10a, not 39/2 a. This is an arithmetic sequence with a common difference of -1/2 a.

In Choice (4), half of 22a is 11a, not 22.5a. This is an arithmetic sequence with a common difference of 1/2 a, or .5a.

8. The result of dividing 2x³ + 6x² + 7x + 2 by x + 1 is

(1) 2² + 4x + 3 - 1 / (x + 1)
(2) 2² + 4x + 3 + 1 / (x + 1)
(3) 2² + 8x - 15 + 17 / (x + 1)
(4) 2² + 8x + 15 - 13 / (x + 1)

Answer: (1) 2² + 4x + 3 - 1 / (x + 1)

If 2x³ + 6x² + 7x + 2 were divisible by (x + 1), then f(-1) would be equal to 0. However, f(-1) = -1, which is a remainder.

Only one choice has -1 / (x + 1) in it, so that is the answer.

Doing it the long way:

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, August 2025 Part IV)

This exam was adminstered in August 2025.

August 2025 Geometry, Part IV

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

35. In quadrilateral ABCD below, side CD is extended through D to point E such that AFD and BFE bisect each other, and DE ≅ DC.

Prove ABCD is a parallelogram.

Answer:

The rule you need to remember here is that a quadrilateral with one pair of opposite sides that are both parallel and congruent is a parallelogram.

We can prove this by showing that the two triangles in the image are congruent. This will show that the AB || CD and then the transitive property will show that AB ≅ CD.

Statement	Reason
1. quadrilateral ABCD with side CD is extended through D to point E such that AFD and BFE bisect each other, and DE ≅ DC.	Given.
2. AF ≅ FD and EF ≅ FB.	Definition of bisector.
3. ∠AFB ≅ ∠DFE	Vertical angles are congruent.
4. △DFE ≅ △AFB	SAS Postulate
5. ∠DEF ≅ ∠ABF	CPCTC
6. AB || EC	If two lines are cut by a transversal and the Alternate Interior Angles are congruent, then the lines are paralllel.
7. AB ≅ DE	CPCTC
8. AB ≅ DC	Transitive Property / Substitution
9. ABCD is a parallelogram	In a quadrilateral, if two opposite sides are parallel and congruent then the quadrilateral in a parallelogram.

One curious note:
In Step 8, I would've stated the reason as "the transitive property". It's curious that in the all the correct solutions in the model response set, the word "substitution" is used. In fact, there's only one example that uses the phrase "Transitive Property", but it uses it incorrectly -- there was a step missing before that. This has me questioning if that is a sufficient "Reason" for full credit. I like it better than "substitution" myself, which is also logical, but sounds more algebraic to me.

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

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

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

This exam was adminstered in August 2025.

August 2025 Geometry, Part III

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

32. Joan wants to fill an empty 75-liter fish tank with water. She uses a cylindrical bucket with a diameter of 20 cm.

Determine and state the maximum number of buckets of water, filled to an exact height of 26 cm, Joan can put into the fish tank before it overflows.

[1000 cm³ = 1 liter]

Answer:

Find the Volume of the bucket. Then divide 75,000 cm³ by the Volume of the bucket to find out how many buckets you'll need.

V = π r²h = π (10)² (26) = 8168.14...

75000 / 8168.14 = 9.18

She can fill nine buckets before it would overflow.

33. As modeled in the diagram below, two cables are attached from a point on a tree 12 feet above the ground. The longer cable is anchored on the ground 3 feet farther from the tree than the shorter cable is anchored. The angle of elevation between the shorter cable and the ground is 50°.

Determine and state, to the nearest foot, the distance from the base of the tree to the point where the longer cable is attached to the ground.

Determine and state, to the nearest degree, the angle of elevation between the longer cable and the ground.

Answer:

Use Tangent to find the length of the base. You have the opposite, and you want the adjacent side, so that is tangent.

Tan 50 = 12/x

x = 12 / tan 50 = 10.069.

Add the extra 3 feet and the base is 13.069, or 13 feet, to the nearest foot.

Next, to find the angle, you have the opposite and the adjacent, so you have to use the inverse of the tangent ratio.

Tan A = 12 / 13

A = tan^-1 (12/13) = 42.7 degrees.

To the nearest angle, the answer is 43 degrees.

34. Quadrilateral READ has vertices with coordinates R(–1,3), E(2,7), A(10,1), and D(7,–3). Prove READ is a rectangle.

[The use of the set of axes below is optional.]

Answer:

Graphing may make this easier, but you still need to explain everything. The graph is part of your justification but it isn't an answer on its own. You can also use formulas and ignore the grid.

A rectangle is a parallelogram with at least one right angle. (If one angle in a paralllelogram is a right angle, then all four must be.)

This means that you can find the slopes of the four lines. Show that the opposite sides are parallel and that one pair of consecutive sides are perpendicular.

Slope of RE = (7 - 3)/(2 - -1) = 4/3.

Slope of EA = (1 - 7)/(10 - 2) = -6/8 = -3/4.

Slope of AD = (-3 - 1)/(7 - 10) = -4/-3 = 4/3.

Slope of DR = (3 - -3)/(-1 - 7) = 6/-8 = -3/4

RE || AD because they have the same slope.

EA || DR because they have the same slope.

RE ⊥ EA because the products of its slopes is -1: (4/3)(-3/4) = -1. (The slopes are negative reciprocals.)

You could answer show that the opposite sides had the same length, but you'd still need to prove that there were perpendicular lines.

You could also show that the diagonals are congruent to each other, but first you'd need to prove that it was a parallelogram.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
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Geometry Problems of the Day (Geometry Regents, August 2025 Part II)

This exam was adminstered in August 2025.

August 2025 Geometry, Part II

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

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

Answer:

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

That's all you need to write.

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

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

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

Answer:

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

V = (5)³ = 125 cm³.

d = 982.5 / 125 = 7.86

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

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

Answer:

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

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

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

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

Point A is located at (3,4).

Another formula you use is the following:

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

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

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

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

Answer:

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

sin (4.8°) = 0.6 / x

x = 0.6 / sin (4.8°) = 7.17...

The ramp has a length of about 7.2.

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

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

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

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

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

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

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

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

(x / 360) π(5)² = 7.5π

(x / 360) 25π = 7.5π

x = (360) (7.5π) / (25π)

x = 108

The measure of KLM is 108 degrees.

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

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

End of Part II

How did you do?

Questions, comments and corrections welcome.

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