Saturday, October 23, 2021

Geometry Problems of the Day (Geometry Regents, June 2012)



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, June 2012

Part I: Each correct answer will receive 2 credits.


21. In circle O shown below, diameter DB is perpendicular to chord AC at E.


If DB = 34, AC = 30, and DE > BE, what is the length of BE?

1) 8
2) 9
3) 16
4) 25

Answer: 2) 9


The first thing you should do is draw OA and OC. These are both radii with a length of 17, which is half of 34. Since the chord is perpendicular to the diameter, then OEA and OEC are both right triangles, each with a leg of 15 and a hypotenuse of 17.

If you didn't recognize the Pythagorean Triple, then you can work out

(OE)2 + 152 = 172

which will tell you that OE = 8 because 8-15-17 is a triple.

However, the question didn't ask for OE. It asked for BE. BE = OB - OE = 17 - 8 = 9.





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


Which statement must be true?

1) AC ≅ DB
2) ∠ABD ≅ ∠CBD
3) triangle AED ≅ triangle ∠CED
4) triangle DCE ≅ triangle ∠BCE

Answer:3) triangle AED ≅ triangle ∠CED


The diagonals of a parallelogram bisect each other. They are NOT congruent unless you know that the parallelogram is also a rectangle. You can prove using SSS that the opposite triangles are congruent -- that is, the ones that share an entire diagonal or the ones on the opposites sides of the vertex E.

Choice (1) MAY be true if ABCD is a rectangle, but it is NOT true otherwise. Eliminate Choice (1).

Choice (2) MAY be true if ABCD is a rhombus where the diagonals are angle bisectors, but it is NOT true otherwise. Eliminate Choice (2).

Choice (3) can be shown to be true using SSS -- the opposite sides of the parallelogram are congruents and the diagonals bisect each other. This is the answer.

Choice (4) cannot be shown to be true with the information given. It would only be true for a rhombus.





23. Which equation of a circle will have a graph that lies entirely in the first quadrant?

1) (x - 4)2 + (y - 5)2 = 9
2) (x + 4)2 + (y + 5)2 = 9
3) (x + 4)2 + (y + 5)2 = 25
4) (x - 5)2 + (y - 4)2 = 25

Answer: 1) (x - 4)2 + (y - 5)2 = 9


They changed up the question a little bit. It's nice to see some variety.

For the circle to be in the first quadrant, it would have to have a center (h,k) in the first quadrant, and the radius would have to be smaller than either h or k.

Choice (1) has a center at (4,5) because the signs are flipped. It has a radius of 3. This means that it has no points to the left of (1,5) or below (4,2). This is the solution.

Choice (2) has its center in Quadrant III at point (-4, -5).

Choice (3) has its center in Quadrant III at point (-4, -5).

Choice (4) has its center at (5,4) but the radius is 5. That means that (5,-1) is on the circle, and that is in Quadrant IV.





24. In the diagram below, triangle ABC ∼ triangle RST.


Which statement is not true?

1) ∠A ≅ ∠R
2) AB/RS = BC/ST
3) AB/BC = ST/RS
4) (AB + BC + AC) / (RS + ST + RT) = AB/RS

Answer: 3) AB/BC = ST/RS


The corresponding angles of similar triangles are congruent. The corresponding sides over of similar triangles are proportional to the scale factor.

Choice (1) says that two corresponding angles are congruent. This is true. Eliminate (1).

Choice (2) shows a proportion of two pairs of corresponding sides. This is true. Eliminate (2).

Choice (3) has a incorrect proportion. The second ratio should be RS/ST, not ST/RS. This is the answer.

Choice (4) shows that the ratio of the two perimeters is the same as the ratio of two corresponding sides, which is the scale factor. This is true. Eliminate (4).

If you need to, you could convert to slope-intercept form:

20x - 2y = 6
-2y = -20x + 6
y = 10x - 3

The slope of a line perpendicular to a line with a slope of 10 would be -1/10.




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 2 Problems of the Day (Algebra 2 Regents, June 2012)



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

More Regents problems.

Algebra 2/Trigonometry Regents, June 2012

Part I: Each correct answer will receive 2 credits.


20. The table below displays the results of a survey regarding the number of pets each student in a class has. The average number of pets per student in this class is 2.


What is the value of k for this table?

1) 9
2) 2
3) 8
4) 4

Answer: 4) 4


There are 22 + k students in the room. They have a total of 6 + 20 + 10 + 4k = 36 + 4k pets.

According to the problem, (36 + 4k) / (22 + k) = 2

(36 + 4k) / (22 + k) = 2
(36 + 4k) = 2(22 + k)
36 + 4k = 2k + 44
2k = 8
k = 4

You could also solve this by plugging in the values for k to see which one works.





22. How many negative solutions to the equation 2x3 - 4x2 + 3x - 1 = 0 exist

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

Answer: 4) 0


If you graph y = 2x3 - 4x2 + 3x - 1, you will see that the curve only has one real root at x = 1. The other two roots are imaginary.

There are no negative roots.





23. A study finds that 80% of the local high school students text while doing homework. Ten students are selected at random from the local high school. Which expression would be part of the process used to determine the probability that, at most, 7 of the 10 students text while doing homework?

1) 10C6 (4/5)6 (1/5)4
2) 10C7 (4/5)10 (1/5)7
3) 10C8 (7/10)10 (3/10)2
4) 10C9 (7/10)9 (3/10)1

Answer: 1) 10C6 (4/5)6 (1/5)4


Part of the process means that it could be anything from the probability that 0 of 10 through 7 out of 10.

Even if you weren't familiar with how to set up the expression, there are hints in the other choices that allow you to eliminate them:

In Choice (2), the exponents are 10 and 7, but they should add up to 10.

In Choice (3), the exponents are 10 and 2, but they should add up to 10.

In Choice (4), the exponents are 9 and 1, which add up to 10, but the fractions are incorrect. The 80% is 4/5, not 7/10. Likewise, if we wanted at most 7, the first part wouldn't be 10C9 because we don't want 9 out of 10.

Choice (1) shows the probability of exactly 6 out of 10 text while doing homework, which is one part of the process.





24. In which interval of f(x) = cos(x) is the inverse also a function?

1) -π/2 < x < π/2
2) -π/2 < x < π/2
3) 0 < x < π
4) π/2 < x < 3π/2

Answer: 3) 0 < x < π


Between 0 and π, the value of f(x) is decreasing from 1 to -1. In that interval, no value repeats for f(x).

For the other intervals, the curve has either a minimum or maximum value, and the values for f(x) will repeat. You could draw a horizontal line through the graph and hit multiple points.




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.



Friday, October 22, 2021

Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)



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

More Regents problems.

Algebra 2/Trigonometry Regents, June 2012

Part I: Each correct answer will receive 2 credits.


16. Which expression is equivalent to (n°m°p)(x), given m(x) = sin x, n(x) = 3x, and p(x) = x2?

1) sin(3x)2
2) 3 sin x2
3) sin2(3x)
4) 3 sin2x

Answer: 2) 3 sin x2


As with compositions of transformations, when you have a composition of functions, you work from right to left. You want to find n OF m OF p OF x, or more explicitly, n(m(p(x))).

p(x) = x2

m(x2) = sin x2

n(sin x2) = 3 sin x2

The other choices will occur if you do operations in the wrong order.





17. The value of csc 138°23' rounded to four decimal places is

1) -1.3376
2) -1.3408
3) 1.5012
4) 1.5057

Answer: 4) 1.5057


If you can't calculate that directly in your calculator, remember that CSC is 1/SIN.

Since Sin is positive between 0 and 180°, you can eliminate Choices (1) and (2). (Without checking, my guess is that both of these are related to Cosine.)

23/60 = .383333333

1/SIN(138.38333333) = 1/0.664143715 = 1.50569821 = 1.5057





18. Which function is one-to-one?

1) k(x) = x2 + 2
2) g(x) = x3 + 2
3) f(x) = |x| + 2
4) j(x) = x4 + 2

Answer: 2) g(x) = x3 + 2


For a relation to be a function, every x can have only 1 value associated with it. It must pass the Vertical Line Test.

For a function to be one-to-one, every y value must be associate with exactly 1 x value. It must pass a horizontal line test as well.

Choices (1), (3) and (4) are all U or V shaped. They come down from infinity, hit a minimum point and turn back up toward infinity. Every y value, except the minimum, will be associated with both a positive AND a negative x value.





19. The conjugate of the complex expression -5x + 4i is

1) 5x - 4i
2) 5x + 4i
3) -5x - 4i
4) -5x + 4i

Answer: 3) -5x - 4i


When it came to binomials, and the Difference of Squares rule, (x + y)(x - y) were conjugates. The two terms were the same but the sign in the middle was change from + to -.

With complex numbers, it is the same thing: (-5x + 4i) and (-5x - 4i) are conjugates. The sign in the middle -- the sign before the imaginary part of the complex number -- changes. The sign of the Real portion does not change.





20. What is a positive value of tan 1/2 x, when sin x = 0.8?

1) 0.5
2) 0.4
3) 0.33
4) 0.25

Answer: 1) 0.5


The value of of sin x = 0.8 when x = 51.130... degrees, as in a 6-8-10 right triangle.

Half of 51.130... is 26.565..

Tan 25.565 = 0.478..., which rounds to .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.



Geometry Problems of the Day (Geometry Regents, June 2012)



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, June 2012

Part I: Each correct answer will receive 2 credits.


16. In the diagram of ABC shown below, DE || BC.


If AB = 10, AD = 8, and AE = 12, what is the length of EC?

1) 6
2) 2
3) 3
4) 15

Answer: 3) 3


Set up a proportion to find the size of AC and subtract the length of AE to get EC.

8 / 10 = 12 / AC
8 AC = 120
AC = 15

EC = AC - AE = 15 - 12 = 3





17. What is the length of AB with endpoints A(-1,0) and B(4,-3)?

1) √(6)
2) √(18)
3) √(34)
4) √(50)

Answer: 3) √(34)


Use the Distance Formula or the Pythagorean Theorem.

√( (-1 - 4)2 + (0 - -3)2 ) = √( (-5)2 + (3)2 ) = √(34)





18. The sum of the interior angles of a polygon of n sides is

1) 360
2) 360/n
3) (n - 2) • 180
4) (n - 2) • 180 / n

Answer: 3) (n - 2) • 180


This is basically a definition question. This is the formula for the sum of the interior angles -- the number of triangles you can make times 180. The number of triangles you can make is Two less than the Number of sides that the polygon has.

Choice (1) is the total of the exterior angles. It's always 360, regardless of the number of sides.

Choice (2) is the size of each exterior angle in a regular polygon.

Choice (4) is the size of each interior angle in a regular polygon.





19. What is the slope of a line perpendicular to the line whose equation is 20x - 2y = 6?

1) -10
2) -1/10
3) 10
4) 1/10

Answer: 2) -1/10


The slope of a line perpendicular to the given line will be the inverse reciprocal of the slope of the given line.

The slope of a line written in standard form, Ax + By = C, is -A/B, which in this case is -(20)/(-2) = 10.

If you need to, you could convert to slope-intercept form:

20x - 2y = 6
-2y = -20x + 6
y = 10x - 3

The slope of a line perpendicular to a line with a slope of 10 would be -1/10.





20. Which graph represents a circle whose equation is (x + 2)2 + y2 = 16?

Answer: 3) (see image)


The equation of a circle is given by the formula (x - h)2 + (y - k)2 = r2, where (h,k) in the center of the circle and r is the radius. Note that there are MINUS signs in the formula, so the signs will be flipped.

Note that this comes up so often that I cut and paste the above sentence from blog post to blog post.

The radius of the circle is √(16) = 4. Eliminate Choices (1) and (4), which have radius 8 and diameter 16.

The center of the circle, according to the equation, is (-2, 0) because the sign if flipped. That's Choice (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.



Math Horror Movies: Too Gross!

(Click on the comic if you can't see the full image.)
(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

You'll get dozens and dozens and dozens and dozens of complaints! And More!

It's just too much! And you know just how much!!



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.



Thursday, October 21, 2021

Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)



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

More Regents problems.

Algebra 2/Trigonometry Regents, June 2012

Part I: Each correct answer will receive 2 credits.


11. Which graph represents the function log 2 x = y? position?


Answer: 1) (see image)


The log2 x function is the inverse of the exponential function 2x, a reflection across the line y = x.

When x = 1 then y = 0 because when 0 is the exponent of 2 then the result is 1.





12. A circle is drawn to represent a pizza with a 12 inch diameter. The circle is cut into eight congruent pieces. What is the length of the outer edge of any one piece of this circle?

1) 3π/4
2) π
3) 3π/2
4) 3π

Answer: 3) 3π/2


The circumference of the pizza is 12π. One eighth of that is 12π/8, which simplifies to 3π/2.





13. What is the solution set for the equation √(5x + 29) = x + 3?

1) {4}
2) {-5}
3) {4, 5}
4) {-5, 4}

Answer: 1) {4}


You can work this out (see below) or you can check if 4, -5 or 5 work.

√(5x + 29) ?= x + 3
√(5(4) + 29) ?= (4) + 3
√(49) ?= 7
True, x = 4 is a solution. Eliminate Choice (2).

√(5x + 29) ?= x + 3
√(5(5) + 29) ?= (5) + 3
√(54) ?= 8
False, x = 5 is NOT a solution. Eliminate Choice (3).

√(5x + 29) ?= x + 3
√(5(-5) + 29) ?= (-5) + 3
√(4) ?= -2
False, x = 2 is NOT a solution. Eliminate Choice (4).

Choice (1) is the solution.

Alternatively, you could solve the equation:

√(5x + 29) = x + 3
5x + 29 = (x + 3)2
5x + 29 = x2 + 6x + 9
x2 + x - 20 = 0
(x + 5)(x - 4) = 0
x + 5 = 0 or x - 4 = 0
x = -5 or x = 4

Next check for extraneous values. As shown above x = -5 is NOT a solution. Discard it. Then x = 4 is the only solution.





14. When factored completely, x3 + 3x2 - 4x - 12 equals

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

Answer: 2) (x + 2)(x - 2)(x + 3)


It says "factored completely" and (x2 - 4) is not factored completely. It can be broken down further. Eliminate Choices (3) and (4).

If you factor the polynomial by grouping:

x3 + 3x2 - 4x - 12
= x3 - 4x + 3x2 - 12
= x(x2 - 4) + 3(x - 4)
= (x2 - 4)(x + 3)
= (x + 2)(x - 2)(x + 3)





15. What is the middle term in the expansion of (x/2 - 2y)6?

1) 20x3y3
2) -15/4 x4y2
3) -20x3y3
4) 15/4 x4y2

Answer: 3) -20x3y3


The expansion has 7 terms, starting with the x6y0 term and ending with the x0y6 term. The middle term would be the fourth one and it would be in the form x3y3, so eliminate Choices (2) and (4).

The coefficient of the fourth term would be 6C3 * (1/2)3 * (-2)3 = (20)(1/8)(-1/8) = -20.




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, June 2012)



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, June 2012

Part I: Each correct answer will receive 2 credits.


11. In the diagram of triagnle ABC shown below, D is the midpoint of AB, E is the midpoint of BC, and F is the midpoint of AC.


If AB = 20, BC = 12, and AC = 16, what is the perimeter of trapezoid ABEF?

1) 24
2) 36
3) 40
4) 44

Answer: 4) 44


Add all of AB to half of AC and half of BC. Then add half of AB, which is the length of EF. So 20 + 6 + 8 + 10 = 44

Choice (1) is the size of triangle DEF which connects the midpoints.

Choices (2) and (3) are the perimeters of the other two trapezoids.





12. In the diagram below, LMO is isosceles with LO = MO.


If m∠L = 55 and m∠NOM = 28, what is m∠N?

1) 27
2) 28
3) 42
4) 70

Answer: 1) 27


Since m∠L = 55 then m∠LMO = 55. This makes m∠OMN = 125 becuase it is supplementary.

Triangle MNO has 180 degrees (like all triangles), so m∠N = 180 - 28 - 125 = 27

Choice (2) is there in case you thought MNO was isosceles. There is nothing that suggests that it is (because it is NOT).





13. If AB is contained in plane P, and AB is perpendicular to plane R, which statement is true?

1) AB is paralel to plane R.
2) Plane P is parallel to plane R.
3) AB is perpendicular to plane P.
4) Plane P is perpendicular to plane R.

Answer: 4) Plane P is perpendicular to plane R.


AB is in P. AB is ⊥ R. Therefore P ⊥ R.

It's that simple.

If it were more complicated than that, you would need a lot more information to make a decision. But it isn't and you don't.





14. In the diagram below of triangle ABC, AE ≅ BE, AF ≅ CF, and CD ≅ BD.


Point P must be the

1) centroid
2) circumcenter
3) incenter
4) ortocenter

Answer: 1) centroid


Points D, E, and F are all midpoints. That makes AB, BC, and CA medians. Medians meet at the centroid.

The circumcenter is the concurrence point for three perpendicular bisectors of a triangle. AD, BF, and CE are NOT shown to be perpendicular to the lines they bisect.

The incenter is the concurrence point for three angle bisectors of a triangle.

The orthocenter is the concurrence point for three altitudes of a triangle.





15. What is the equation of the line that passes through the point (-9,6) and is perpendicular to the line y = 3x - 5?

1) y = 3x + 21
2) y = -1/3 x - 3
3) y = 3x + 33
4) y = -1/3 x + 3

Answer: 4) y = -1/3 x + 3


Perpendicular lines have slopes that are inverse reciprocals. Parallel lines have the same slopes.

The given line has a slope of 3. Choices (1) and (3) are lines with a slope of 3, so they are parallel to the given line. Eliminate them.

The slope of the perpendicular line must be -1/3. Check (-9,6) to see if it is a point on either line.

y = -1/3(-9) - 3 = 3 - 3 = 0. (-9, 6) is not on this line.

y = -1/3(-9) + 3 = 3 + 3 = 6. (-9, 6) IS a point on this line.




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.



Wednesday, October 20, 2021

Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)



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

More Regents problems.

Algebra 2/Trigonometry Regents, June 2012

Part I: Each correct answer will receive 2 credits.


6. If m∠θ = -50, which diagram represents θ drawn in standard position?


Answer: 4) (see image)


If θ has a negative value, you drop down below the x-axis, in a clockwise direction, starting from the right side of axis.

Choice (3) is 50 degrees. Choices (1) and (2) are not standard positions because they start on the left.





7. If logb x = 3 log b p - (2 log b t + 1/2 log b r), then the value of x is


Answer: 4) See image


Sometimes I think the Regents people just want to give me a headache.





8. Which equation has roots with the sum equal to 9/4 and the product equal to 3/4?

1) 4x2 + 9x + 3 = 0
2) 4x2 + 9x - 3 = 0
3) 4x2 - 9x + 3 = 0
4) 4x2 - 9x - 3 = 0

Answer: 3) 4x2 - 9x + 3 = 0


The rule for the Sum of the Roots is −b/a, and for the Product of the Roots is c/a. Note that the Sum has a minus sign and the Product does not.

If -b/a = +9/4 then either a or b must be negative. Since a = 4 in all four choices, b must be -9. Eliminate Choices (1) and (2).

If c/a = +3/4 then either a and c are both positive or they are both negative. Since a must be +4, then c must be +3. Eliminate Choice (4).





9. Which graph represents the solution set of | (4x - 5) / 3 | > 1?

Answer: 3) see image


You have to remove the absolute value symbols and then solve TWO inequalities: one for > 1 and one for < -1

(4x - 5) / 3 > 1
4x - 5 > 3
4x > 8
x > 2
(4x - 5) / 3 < -1
4x - 5 < -3
4x < 2
x < 1/2





10. Which expression is equivalent to (x-1y4) / (3x-5y-1)

1) x4y5 / 3
2) x5y4 / 3
3) 3x4y5
4) y4 / (3x5)

Answer: 1) x4y5 / 3


When you have the same variable in the numerator and the denominator, you can SUBTRACT the exponents.

x-1 / x-5 = x-1 - -5 = x4

y4 / y-1 = y4 - -1 = y5

The 3 in the denominator doesn't move. It remains 1/3 because there is no exponent affecting it.

The final result is 1/3 x4 y5




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, June 2012)



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, June 2012

Part I: Each correct answer will receive 2 credits.


6. In the diagram below of ABCD, AC ≅ BD.


Using this information, it could be proven that

1) BC = AB
2) AB = CD
3) AD - BC = CD
4) AB + CD = AD

Answer: 2) AB = CD


Since BC ≅ BC, it could be removed from AC and BD, which are congruent, and the remaining segments will still be congruent because of the Subtraction Property of Congruency.

In Choice (1), there is nothing to suggest that B is a midpoint of AC.

Choice (3) is an incorrect equation. AD - BC = CD PLUS AB.

Likewise, Choice (4) is an incorrect equation. AB + CD = AD MINUS BC.





7. The diameter of a sphere is 15 inches. What is the volume of the sphere, to the nearest tenth of a cubic inch?

1) 706.9
2) 1767.1
3) 2827.4
4) 14,137.2

Answer: 2) 4 is not an odd integer; true


If the diameter is 15 then the RADIUS is 7.5, and you need the radius for the formula, which is 4/3 π r3

(4/3) π (7.5)3 = 1767.14..., or 1767.1, which is Choice (2).

If you used the diameter of 15 by mistake, you would have gotten Choice (4) and wouldn't have known you got it wrong.

Choice (1) shows the SURFACE AREA of a sphere with a 15 inch diameter.

Choice (3) showns the SURFACE AREA of a sphere with a 15 inch RADIUS.





8. The diagram below shows the construction of AB through point P parallel to CD.


Which theorem justifies this method of construction?

1) If two lines in a plane are perpendicular to a transversal at different points, then the lines are parallel.
2) If two lines in a plane are cut by a transversal to form congruent corresponding angles, then the lines are parallel.
3) If two lines in a plane are cut by a transversal to form congruent alternate interior angles, then the lines are parallel.
4) If two lines in a plane are cut by a transversal to form congruent alternate exterior angles, then the lines are parallel.

Answer: 2) If two lines in a plane are cut by a transversal to form congruent corresponding angles, then the lines are parallel.


Knowing nothing else about construction, you can deduce that the answer is Choice (2) because the angles that are marked off are corresponding angles.

The angles were not constructed to be perpendicular. The are neither alternate interior nor exterior. Eliminate Choices (1), (3) and (4).

The construction shown is how you can draw a line parallel to a given line through a point, P, not on that line. The theorem about corresponding angles justifies this.





9. Parallelogram ABCD has coordinates A(1,5), B(6,3), C(3,-1), and D(-2,1). What are the coordinates of E, the intersection of diagonals AC and BD?

1) (2,2)
2) (4.5,1)
3) (3.5,2)
4) (-1,3)

Answer: 1) (2,2)


THe diagonals of a parallelogram bisect each other. You need to find the midpoint of AC or of BD. Or you can check your work by finding both of them, which MUST BE the same point.

To find the midpoint, take the average of the two x-coordinates and the average of the two y-coordinates.

E = ( (1+3)/2, (5-1)/2 ) = (2,2)

E = ( (6-2)/2, (3+1)/2 ) = (2,2)

Choice (3) is the midpoint of AB. I don't see the "logic" behind the other two incorrect choices.





10. What is the equation of a circle whose center is 4 units above the origin in the coordinate plane and whose radius is 6?

1) x2 + (y - 6)2 = 16
2) (x - 6)2 + y2 = 16
3) x2 + (y - 4)2 = 36
4) (x - 4)2 + y2 = 36

Answer: 3) x2 + (y - 4)2 = 36


The equation of a circle is given by the formula (x - h)2 + (y - k)2 = r2, where (h,k) in the center of the circle and r is the radius. Note that there are MINUS signs in the formula, so the signs will be flipped.

Note that this comes up so often that I cut and paste the above sentence from blog post to blog post.

The radius is 6, so r2 = 36. Eliminate choices (1) and (2).

Four units above the origin is the point (0,4), which is the center. That means that the correct equation is Choice (3).

Choice (4) would have been correct if the center was four units to the RIGHT of the origin, at (4,0).




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.



Tuesday, October 19, 2021

Geometry Problems of the Day (Geometry Regents, June 2012)



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, June 2012

Part I: Each correct answer will receive 2 credits.


1. Triangle ABC is graphed on the set of axes below.


Which transformation produces an image that is similar to, but not congruent to, triangle ABC?

1) T2,3
2) D2
3) ry = x
4) R90

Answer: 2) D2


A Dilation does not preserve distance, or size, by defintion. The shape will be conserved, meaning that all of the angles in the image are congruent to the original, which is the definition of similar.

Translations, reflections and rotations are produce images that are congruent to the original pre-image.





2. A student wrote the sentence “4 is an odd integer.” What is the negation of this sentence and the truth value of the negation?

1) 3 is an odd integer; true
2) 4 is not an odd integer; true
3) 4 is not an even integer; false
4) 4 is an even integer; false

Answer: 2) 4 is not an odd integer; true


The negation of "is" is "is not", and the rest of the sentence doesn't matter.

The negation of this sentence is "4 is not an odd integer" and that negation is true.

Negative does not make 4 into 3. Why 3? Why not any other number? This makes no sense. A negation is very specific.

Changing "is an odd" to "is not an even" requires changing TWO things: adding the "not" and switching "odd" to "even". This is not how a negation is done.

With Choice (4), it could be argued argued that "not an odd integer" is the same as "an even integer". However, this ignores the case that it might not be an integer at all, which would mean that the number is neither odd nor even. There are more than two possibilities. In any event, the statment "4 is an even integer" is true, not false, so the Choice is incorrect regardless of any argument.





3. As shown in the diagram below, EF intersects planes P , Q , and R/


If EF is perpendicular to planes P and R , which statement must be true?

1) Plane P is perpendicular to plane Q.
2) Plane R is perpendicular to plane P.
3) Plane P is parallel to plane Q.
4) Plane R is parallel to plane P.

Answer: 4) Plane R is parallel to plane P.


If you picture EF as a horizontal line, it would be perpendicular to vertical planes. That would make planes P and R parallel to each other.

Plane Q has no special relationship with either planes P or R, nor with line EF. Plane Q is shown to intersect P and R, so it cannot be parallel to them. If it were perpendicular to P and R, then plane Q would either have to be parallel to EF, which is can't because they intersect, or it would have to contain the line EF in its entirety. It isn't shown to do that, and the word "intersect" would suggest that this isn't the case.





4. In the diagram below, LATE is an isosceles trapezoid with LE ≅ AT, LA = 24, ET = 40, and AT = 10. Altitudes LF and AG are drawn.


What is the length of LF?

1) 6
2) 8
3) 3
4) 4

Answer: 1) 6


Since it is an isosceles trapezoid, you know that triangles LEF and AGT are congruent (by AAS, if you need to work it out). That means that EF ≅ GT. Call each of those x.

Also, FG ≅ LA.

So x + 24 + x = 40, meaing 2x + 24 = 40. Then x = 8.

So EF = 8, LE = 10 and LFE is a right triangle.

A quick use of Pythagorean Theorem tells you that LF = 6. And you should seriously know 6-8-10 is a Pythagorean Triple without needing to use the theorem.

remain parallel.





5. In the diagram below of circle O, diameter AB is parallel to chord CD.


If mCD = 70, what is mAC?

1) 110
2) 70
3) 55
4) 35

Answer: 3) 55


AC = BD and AC + CD + BD = 180

So 2AC + 70 = 180
2AC = 110
AC = 55




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 2 Problems of the Day (Algebra 2 Regents, June 2012)



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

More Regents problems.

Algebra 2/Trigonometry Regents, June 2012

Part I: Each correct answer will receive 2 credits.


1. What is the product of (2/5 x - 3/4 y2) and (2/5 x + 3/4 y2)?

1) 4/25 x2 - 9/16 y4
2) 4/25 x - 9/16 y2
3) 2/25 x2 - 3/4 y4
4) 4/5 x

Answer: 1) 4/25 x2 - 9/16 y4


The multiplication of two conjugates (the same binomial except one has a plus and the other a minus) will have a product that is the Difference of Two Perfect Squares.

In general: (ax + by) * (ax - by) = a2x2 - b2y2.

Replace a with 2/5 and b with 3/4 and you get:

(2/5 x + 3/4 y) * (2/5 x - 3/4 y) = (2/5)2x2 - (3/4)2y2
= 4/25 x2 - 9/16 y2





2. What is the domain of the function shown below?



1) -1 < x < 6
2) -1 < y < 6
3) 2 < x < 5
4) 2 < y < 5

Answer: 1) -1 < x < 6


Domain refers to the x values, so eliminate Choices (2) and (4) which refer to y.

The only values of x that have points are from -1 through 6, inclusive. That is Choice (1).

Note that Choice (4) is the range of the function.





3. What is the solution set for 2cosθ - 1 = 0 in the interval 0° < θ < 360°

1) {30°, 150°}
2) {60°, 120°}
3) {30°, 330°}
4) {60°, 300°}

Answer: 4) {60°, 300°}


Solve for cosθ, and then find the values of θ between 0 and 360 that will give the correct answer.

2cosθ - 1 = 0
2cosθ = 1
cosθ = 1/2

The values on the Unit Circle when cosθ = 1/2 is at 60 degrees above and below the x-axis. However, since the stated interval was 0 to 360, we want 60 and 300, not 60 and -60.

-5 - -3 = -2 and 2 - 6 = -4

Point B is (-5 - 2, 2 - 4), or (-7, -2)





4. The expression ∛(64a16) is equivalent to

1) 8a4
2) 8a8
3) 4a5 ∛(a)
4) 4a5 ∛(a5)

Answer: 3) length of sides


The cube root of 64 is 4 because 43 = 64.

To find the cube root of a16, break it into a5 * a5 * a5 * a1. So the cube root is a5 * ∛(a)

So the final answer is 4 * a5 * ∛(a)

A dilation will retain the orientation and the shape of the original. Since the shape is similar, the size angles will be the same and lines that were parallel will remain parallel.





5. Which summation represents 5 + 7 + 9 + 11 + ... + 43?

Answer: 2) (see image)


The summations are like little loops (FOR/NEXT, or DO UNTIL, etc, for programmers)

In Choice (1), all the integers from 5 to 43 are being added, but we only want the odd numbers. Eliminate Choice (1).

In Choice (2), the first number is 2(1) + 3 = 5 and the last is 2(20) + 3 = 43. The sequence is counting by 2s. This is the solution.

In Choice (3), the first number is 2(4) - 3 = 5 but the last is 2(24) - 3 = 45, not 43. Had the top number been 23 instead of 24, this would have worked. Eliminate Choice (3).

In Choice (4), the sequence is increasing by 3 and not 2. Eliminate Choice (4).

Once you see that the series is increasing by 2, Choices (1) and (4) should have been eliminated immediately.




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.



Monday, October 18, 2021

Math Horror Movies: Red

(Click on the comic if you can't see the full image.)
(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

In Space, No One Can Hear You Get Mad Dept.

That time of year, again, when I need more shlocky Halloween-type horror B-movie titles.

I love 'em and hate 'em at the same time.



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.