Geometry Problems of the Day (Geometry Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, August 2010

Part I: Each correct answer will receive 2 credits.


11. In △ABC, AB = 7, BC = 8, and AC = 9. Which list has the angles of △ABC in order from smallest to largest?

1) ∠A, ∠B, ∠C
2) ∠B, ∠A, ∠C
3) ∠C, ∠B, ∠A
4) ∠C, ∠A, ∠B

Answer: 4) ∠C, ∠A, ∠B

The sides from smallest to largest are AB = 7, BC = 8, and AC = 9.

Therefore, the angles from smallest to largest are ∠C, ∠A, ∠B, which is Choice (4).

12. Tangents PA and PB are drawn to circle O from an external point, P, and radii OA and OB are drawn. If m∠APB = 40, what is the measure of ∠AOB?

1) 140°
2) 100°
3) 70°
4) 50°

Answer: 1) 140°

∠AOB is supplementary to ∠APB. The two tangents and two radii form a quadrilateral (more specifically, a kite), which has 360°. The tangents meet the radii at right angles, and 90 + 90 = 180.

So 180 - 40 = 140.

13. What is the length of the line segment with endpoints A(−6,4) and B(2,−5)?

1) √13
2) √17
3) √72
4) √145

Answer: 4) √145

Pythagorean Theorem or Distance Formula.

The difference in the x values is 8. The difference in the y values is 9.

√(8² + 9²) = √(64 + 81) = √145, which is Choice (4).

14. The lines represented by the equations y + 1/2 x = 4 and 3x + 6y = 12 are

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

Answer: 2) parallel

Rewrite both equations in the same form, such as slope-intercept.

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

3x + 6y = 12
6y = -3x + 12
y = -1/2x + 2

The two lines have the same slope but different y-intercepts, so the lines are parallel.

15. A transformation of a polygon that always preserves both length and orientation is

1) dilation
2) translation
3) line reflection
4) glide reflection

Answer: 2) translation

A translation preserves length and orientation.

A dilation does not preserve length.

A reflection (line or glide) does not preserve orientation.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, August 2010

Part I: Each correct answer will receive 2 credits.


6. If △ABC ∼ △ZXY, m∠A = 50, and m∠C = 30, what is m∠X?

1) 30
2) 50
3) 80
4) 100

Answer: 4) 100

∠A ≅ ∠Z, ∠B ≅ ∠X and ∠C ≅ ∠Y.

If m∠A = 50 and m∠C = 30, then m∠B = 180 - (50 + 30) = 100

So m∠X = 100.

7. In the diagram below of △AGE and △OLD, ∠GAE ≅ ∠LOD, and AE ≅ OD.

To prove that △AGE and △OLD are congruent by SAS, what other information is needed?

1) GE ≅ LD
2) AG ≅ OL
3) ∠AGE ≅ ∠OLD
4) ∠AEG ≅ ∠ODL

Answer: 2) AG ≅ OL

You are given a pair of congruent sides and a pair of congruent angles. You need the pair of sides that causes the angle to be included between the sides.

That is to say, you need to know that AG and OL are congruent. This is Choice (2).

Choice (1) would give you SSA, which is not a postulate or theorem for congruency.

Choice (3) would give you AAS, instead of SAS.

Choice (4) would give you ASA, instead of SAS.

8. Point A is not contained in plane B. How many lines can be drawn through point A that will be perpendicular to plane B?

1) one
2) two
3) zero
4) infinite

Answer: 1) one

There is only one line from a point to a plane that is pendicular to that plane.

9. The equation of a circle is x² + (y − 7)² = 16. What are the center and radius of the circle?

1) center = (0,7); radius = 4
2) center = (0,7); radius = 16
3) center = (0,−7); radius = 4
4) center = (0,−7); radius = 16

Answer: 1) center = (0,7); radius = 4

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

This means that the center of the circle is (0,7). Eliminate Choices (3) and (4).

The radius is not 16, but &sqrt;(16) = 4. Eliminate Choices (2) and (4).

Choice (1) is the correct choice.

10. What is an equation of the line that passes through the point (7,3) and is parallel to the line 4x + 2y = 10?

1) y = 1/2 x - 1/2
2) y = -1/2 x + 13/2
3) y = 2x − 11
4) y = −2x + 17

Answer: 4) y = −2x + 17

Parallel lines have the same slope. The slope of 4x + 2y = 10 is -A/B, or -4/2 = -2.

The only possibility is Choice (4). Double check: y = -2(7) + 17 = -14 + 17 = 3. Check!

If you didn't remember -A/B to find the slope of a line in Standard Form, you could rewrite the equation:

4x + 2y = 10

2y = -4x + 10

y = -2x + 5

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Complement of a Set

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Complement of a Set and Subset of another Set.

Three comics in three days. I can't promise I'll be this productive for a prolonged time, but I'll keep it going as long as I can. I don't like doing only a single comic per week.

Thanks for all the support from my comic readers.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Come back often for more funny math and geeky comics.

Geometry Problems of the Day (Geometry Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, August 2010

Part I: Each correct answer will receive 2 credits.


1. In the diagram below, △ABC ≅ △XYZ.

Which two statements identify corresponding congruent parts for these triangles?

1) AB ≅ XY and ∠C ≅ ∠Y
2) AB ≅ YZ and ∠C ≅ ∠X
3) BC ≅ XY and ∠A ≅ ∠Y
4) BC ≅ YZ and ∠A ≅ ∠X

Answer: 4) BC ≅ YZ and ∠A ≅ ∠X

AB ≅ XY and BC ≅ YZ. Eliminate Choices (2) and (3).

∠A ≅ ∠X, ∠B ≅ ∠Y and ∠C ≅ ∠Z. Elimate Choice (1).

This leaves Choice (4), which is the correct answer.

2. A support beam between the floor and ceiling of a house forms a 90° angle with the floor. The builder wants to make sure that the floor and ceiling are parallel. Which angle should the support beam form with the ceiling?

1) 45°
2) 60°
3) 90°
4) 180°

Answer: 3) 90°

I hate to be this way with any math question (NARRATOR: He loves it.), but this is a gimme of a question.

The support beam in a house should form a 90° angle with both the floor and the ceiling (which becomes the next floor).

Strictly speaking in Geometry terms, if two lines are parallel, then the same-side interior angles created by a transversal are supplementary. 90 is supplementary to 90 because 180 - 90 = 90.

3. In the diagram below, the vertices of △DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of △ABC is 36 cm.

What is the length, in centimeters, of EF?

1) 6
2) 12
3) 18
4) 4

Answer: 1) 6

The midsegments are half of the length of the sides of the triangle that they are parallel to.

The perimeter of the equilateral triangle is 36, so each side is 12. So each midsegment is 6, which is Choice (1).

4. What is the solution of the following system of equations?

y = (x + 3)² − 4
y = 2x + 5

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

Answer: 3) (−4,−3) and (0,5)

You can put both of these in your calculator. Or you check each of the choices.

The point (0,-4) is NOT the y-intercept of y = 2x + 5. Eliminate Choice (1).

The point (-4,0) is NOT on y = 2x + 5 because 2(-4) + 5 = -3.

Since (0,5) is the y-intertecept of y = 2x + 5, check it with the quadratic:

y = (0 + 3)² - 4 = 3² - 4 = 9 - 4 = 5 So (0,5) is also a solution to the system of equations. This means that Choice (3) is the correct answer.

Double check point (-4,-3):

y = 2(-4) + 5 = -8 + 5 = -3, check!
y = (-4 + 3)² - 4 = (-1)² - 4 = 1 - 4 = -3, check!

5. One step in a construction uses the endpoints of AB to create arcs with the same radii. The arcs intersect above and below the segment. What is the relationship of AB and the line connecting the points of intersection of these arcs?

1) collinear
2) congruent
3) parallel
4) perpendicular

Answer: 4) perpendicular

These are the instructions for constructing a perpendicular bisector of line segment AB. This is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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 (Integrated Algebra Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, August 2010

Part I: Each correct answer will receive 2 credits.


6. The quotient of (9.2 × 10⁶) and (2.3 × 10²) expressed in scientific notation is

1) 4,000
2) 40,000
3) 4 X 10³
4) 4 X 10⁴

Answer: 4) 4 X 10⁴

Put the expression (9.2 × 10⁶) / (2.3 × 10²) into your calculator, and you will get the result of 40,000. (Note: If you got 400,000,000, then you multiplied by 10² instead of dividing. You didn't use parentheses.)

However, 40,000 is NOT in scientific notation. It is equivalent to 4 X 10⁴, which is Choice (4).

7. In a recent town election, 1,860 people voted for either candidate A or candidate B for the position of supervisor. If candidate A received 55% of the votes, how many votes did candidate B receive?

1) 186
2) 837
3) 1,023
4) 1,805

Answer: 2) 837

If 55% voted for A and the rest voted for B, then 45% of the 1,860 people voted for B.

1860(.45) = 837, which is Choice (2).

8. WWhich expression is equivalent to 121 − x²?

1) (x - 11)(x - 11)
2) (x + 11)(x - 11)
3) (11 - x)(11 + x)
4) (11 - x)(11 - x)

Answer: 3) (11 - x)(11 + x)

The expression is the Difference Bewteen Two Perfect Squares, which can be factored into two conjugates, which is the square root of each of the perfect squares, in order.

Choices (1) and (4) are not conjuagates. These will each have a middle term of -22x.

Choice (2) has a product of x² - 121 instead of 121 - x². Eliminate this.

Choice (3) is the correct answer.

9. Given:

U = {1, 2, 3, 4, 5, 6, 7, 8}
B = {2, 3, 5, 6}
Set B is a subset of set U. What is the complement of set B?

1) { }
2) {2, 3, 5, 6}
3) {1, 4, 7, 8}
4) {1, 2, 3, 4, 5, 6, 7, 8}

Answer: 3) {1, 4, 7, 8}

The complement of B is every element of U that is not in B.

Removing 2, 3, 5, and 6 from U leaves 1, 4, 7, 8. This is Choice (3).

Choice (1) is the Empty (Null) Set. Choice (2) is Set B, not the complement. Choice (4) is U, the Universal Set.

10. Which graph can be used to find the solution of the following system of equations?

y = x² + 2x + 3
2y − 2x = 10

Answer: 1) [SEE IMAGE]

You can enter the first equation (the quadratic equation) into your graphing calculator. The second equation (the linear equation) has to be rewrtten to start with y = to use your calculator.

Without using the calculator, the middle term of the quadratic is "+ 2x" and the leading coefficient is positive 1, so the Axis of Symmetry is x = -1. This eliminates Choices (3) and (4). The linear equation will have a positive slope (-A/B), which eliminates Choice (2).

Choice (1) is correct.

If you needed to rewrite the linear equation:

2y − 2x = 10
2y = 2x + 10
y = x + 5

The correct linear equation is shown in (1) and (3).

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

What Am I?

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

C could be i if r was i/(2π).

I thought about making this with i saying "What am i? but the expressions and equations that readily come to mind don't have r in them, and I wanted the r more than I wanted the i in the comic.

You can say that I went all around that joke before deciding on a formula.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Come back often for more funny math and geeky comics.

Operations

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Trust me. It makes scents.

Given the amount of typos I make with my lesson plans, you'd think I'd've done this one years ago.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Come back often for more funny math and geeky comics.

X - Bar

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

It's the name of the bar, not the solution to a puzzle.

Okay, yes, there was a point when I considered back-engineering the frequency table to make the mean, as well as the mode, equal to 5. It wasn't worth the effort. At a minimum, the second table would have to be eliminated.

No comment on the contents of the beverages. Not everything in a math pub has to be a 7-and-7.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Come back often for more funny math and geeky comics.

Primorial

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Soup's on!

Primorial numbers are something new for me. Similar to factorial, and represented with # instead of !, a primorial number, n#, is the product of all the primes less than or equal to n, with 0# = 1# = 1.

They have their uses in math higher than I tend to cover on this page.

I'd rank it a little double the double factorial, !!, which I encountered a decade ago in an article about probability and Texas Hold-em. In that game, each player has 2 hole cards, and their order doesn't matter. This is also likely the first time in nearly a decade that I've thought about Texas Hold'em. The fad kinda passed for me.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Come back often for more funny math and geeky comics.

Zero Search

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

You 2 can be a 0!

First off, this comic was first published in Madrid in Boletín Nº50 from Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid. I promised them some new comics that I hadn't published here first. The problem is that the comic needed to be about math, not about teachers or geeky stuff, and it couldn't rely on a pun that might not translate. (As it is, the comics are not translated into Spanish. Not be me, at any rate.)

I had the idea for this one for a couple of weeks. When I finally decided to do it, I realized that I didn't like any of the titles I came up with for the talent show. The Next Zero, The Next Big Zero, Top Zero. Obviously, I couldn't include "America" since it was debuting in Spain. Finally, I just fell back on Zero Search, which works well for finding the roots of functions.

Yes, I probably overthought all of this.

For the longtime readers, that Nonnie Tyler ("Holding Out for a Zero") as the third celebrity judge.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Integrated Algebra Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, August 2010

Part I: Each correct answer will receive 2 credits.


1. The school store did a study comparing the cost of a sweatshirt with the number of sweatshirts sold. The price was changed several times and the numbers of sweatshirts sold were recorded. The data are shown in the table below.

Which scatter plot represents the data?

Answer: 3) [See Image]

The x-axis is labeled 5, 10, 15, 20, and 25, which matches the costs of the sweatshirts, not the number sold. Note that there are points where x = 5, x = 10, etc. Eliminate Choices (2) and (4).

The first point in Choice (1) is ($5, 9 sold). That's no data from the table. Eliminate Choice (1).

The points in Choice (3) match up with the data: (10, 9), (25, 6), etc.

2. What is the solution of 3(2m − 1) ≤ 4m + 7?

1) m ≤ 5
2) m ≥ 5
3) m ≤ 4
4) m ≥ 4

Answer: 1) m ≤ 5

Solve the inequality using the Distributive Property and inverse operations.

3(2m − 1) ≤ 4m + 7

6m − 3 ≤ 4m + 7

2m − 3 ≤ 7

2m ≤ 10

m ≤ 5

Which is Choice (1).

3. Which set represents the intersection of sets A, B, and C shown in the diagram below?

1) {3, 4, 5, 6, 7}
2) {2}
3) {2, 3, 4, 5, 6, 7}
4) {1, 2, 3, 4, 5, 6, 7, 8, 9}

Answer: 2) {2}

The intersection of A, B and C is the centermost section which is contained within all three circles.

4. The end of a dog’s leash is attached to the top of a 5-foot-tall fence post, as shown in the diagram below. The dog is 7 feet away from the base of the fence post.

How long is the leash, to the nearest tenth of a foot?

1) 4.9
2) 8.6
3) 9.0
4) 12.0

Answer: 2) 8.6

The Leash is the longest side of a right triangle with legs 5 and 7. It cannot be 4.9 feet because that would be shorter than the other sides. It cannot be 12 feet because 5 - 7 - 12 is not a triangle -- it's a straight line.

You know two sides of a right triangle and you want to know the third side. That's a Pythagoean Theorem problem. (It doesn't involve basic trigonometric ratios.)

5² + 7 ² = c²

25 + 49 = c²

74 = c²

c = 8.6

You have a calculator, so use it. However, once you had 74, you should realize that 74 is between 64 and 81, so c was between 8 and 9. There's only one choice between 8 and 9.

5. What is the slope of the line passing through the points A and B, as shown on the graph below?

1) -3
2) -1/3
3) 3
4) 1/3

Answer: 2) -1/3

Rise/run. It's going down, so it's negative. It has a shallow (not steep) slope, so it's a fraction less between 0 and -1. There's only one choice.

Rise / run = -2 / 6 = -1 / 3

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Keys: An Empirical Study

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

The probabilty of successfully unlocking the door on the first time is very low.

In actuality, the incorrect key fits into the cylinder but doesn't turn. However, the correct key, at times, does not turn on the first attempt leading the user (i.e., me) to believe that it is, in fact, the incorrect of the two keys.

However, this wasn't as easy to illustrate, so I went with non-entry, which doesn't actually happen with similar keys. The bathroom key, for example, is a completely different key and wouldn't fit into the cylinder at all. However, there is no circumstance under which I'd mistake the bathroom key for the classroom key.

I will add in these notes that I believe I saw a similar image years ago about orienting a thumb drive correctly. I will defend this comic by saying first, these involves the selection of one out of two separate keys, as opposed to lining up a USB or picking up the correct end of a paper clip. Second, this actually happens to me on a daily basis. I'm living this comic, making this autobiographical.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out 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.

Come back often for more funny math and geeky comics.