Friday, July 19, 2024

June 2024 Algebra Regents Part III


This exam was adminstered in January 2024 .

June 2024 Algebra, Part III

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

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

y = x2 - 3x - 6
y = x - 1

State the coordinates of all solutions.

Answer:


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

Your graph should look like this one:

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



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

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

State the correlation coefficient to the nearest hundredth.

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

Answer:


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

The equation will be y = -37.57x + 215.67

The correlation coefficient is -.98.

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



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

Answer:


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

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

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



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

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

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

Answer:


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

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

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

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

y - x > 1
y > x + 1

You're graph should look like this:

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

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

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories.
Order the softcover or ebook at Amazon.

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!


Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Thursday, July 18, 2024

June 2024 Algebra Regents Part II


This exam was adminstered in January 2024 .

June 2024 Algebra, Part II

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

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

Answer:


Distribute the 5 and use Inverse Operations.

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

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



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

Answer:


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

g(-3) = (-3)3 + 2(-3)2 - (-3) = -6

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

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

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



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

Answer:


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

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

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



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

Complete the two-way frequency table

Answer:


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

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

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

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

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



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

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

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

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



30. Factor 20x3 - 45x completely.

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

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

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

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

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories.
Order the softcover or ebook at Amazon.

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!


Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Sunday, July 07, 2024

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



This exam was adminstered in June 2024.

More Regents problems.

June 2024 Geometry Regents

Part I

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


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

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

Answer: (3) AC ≅ BD


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

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

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

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

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




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

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

Answer: (1) (-2,2)


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

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

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

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

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

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

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

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




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


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

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

Answer: (3) △AEC ~ △BED


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

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

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

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

You know that all four choices cannot be true.




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

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

Answer: (2) 11


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

Therefore,

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

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

Choice (2) is the correct answer.




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

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

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


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

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

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




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

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

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

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


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

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

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

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

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

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




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

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

Answer: (1) MA ≅ MK


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

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

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

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

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




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

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

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


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

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

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




End of Part I.

How did you do?

Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories.
Order the softcover or ebook at Amazon.

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!


Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Saturday, July 06, 2024

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



This exam was adminstered in June 2024.

More Regents problems.

June 2024 Geometry Regents

Part I

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


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


Which statement is always true?

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

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


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

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

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

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

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

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




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


If m∠C = 42o, what is m∠A?

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

Answer: (1) 24o


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

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

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

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

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




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

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

Answer: (2) 26 π


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

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

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

A = 260/360 (6)2 = 26

The correct answer is Choice (2).




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

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

Answer: (4) 12.3


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

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

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

The correct answer is Choice (4).

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

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




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

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

Answer: (4) 12.3


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

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

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

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




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

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

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


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

The correct choice is Choice (3).

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




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

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

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


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

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

The correct answer is Choice (2).




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

(1) √(48)
(2) √(80)
(3) 12
(4) 16

Answer: (4) 16


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

(8)2 = (4)(x)
64 = 4x
x = 16

The correct answer is Choice (4).




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories.
Order the softcover or ebook at Amazon.

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!


Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Friday, July 05, 2024

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



This exam was adminstered in June 2024.

More Regents problems.

June 2024 Geometry Regents

Part I

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


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


Which statement is always true?

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

Answer: (4) OE ≅ AT


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

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

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

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




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

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

Answer: (2) circle


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

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

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




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

(1) 45°
(2) 72°
(3) 60°
(4) 120°

Answer: (3) 60°


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

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

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





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


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

(1) 108π
(2) 432π
(3) 972π
(4) 7776π

Answer: (3) 972π


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

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

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




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

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

Answer:(1) 2.47


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

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

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

Comparing sides, you will see that

5.4 / x = 7.2 / 3.29

so 7.2 x = (5.4)(3.29)

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

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




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

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

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


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

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

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

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

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

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




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


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

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

Answer: (3) m∠HMR' = 40°


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

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

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

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

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




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

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

Answer: (2) 245


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

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

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

The correct answer is Choice (2).




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories.
Order the softcover or ebook at Amazon.

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!


Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Wednesday, July 03, 2024

June 2024 Geometry Regents Part IV


This exam was adminstered in June 2024 .

June 2024 Geometry, Part IV

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

35. Triangle JOE has vertices whose coordinates are J(4,6), O(-2,4), and E(6,0).
Prove that △JOE is isosceles.
[The use of the set of axes on the next page is optional.]

Point Y(2,2) is on OE.
Prove that JY is the perpendicular bisector of OE.

Answer:


As far as Part IV questions go, this one isn't terrible. In fact, it's pretty simple compared to some of the ones they ask involving parallelograms.

To show that JOE is an isosceles triangle, you have to find the lengths of all three sides, state which two of them are congruent. Then state JOE is an isosceles triangle because it has to congruent sides. (YES! You need to write this concluding statement!)

You can use the Distance Formula, or, if you graph JOE, you can just count boxes and use Pythagorean Theorem.

JO = √(62 + 22) = √(40)

OE = √(82 + 42) = √(80)

EJ = √(22 + 62) = √(40)

Since JO ≅ EJ, triangle JOE is an isosceles triangle.

To prove that JY is the perpendicular bisector of OE, you have to prove that JY perpendicular to OE and that JY is the bisector of OE. Yes, you need to do both things.

If the lines are perpendicular, then the slopes will be inverse reciprocals (that is, they have a product of -1). If JY is a bisector, then it goes through the midpoint of OE.

The midpoint of OE is ( (-2+6)/2, (4+0)/2 ) or (2,2), which is point Y. So Y is the midpoint of OE and JY is a bisector of OE.

The slope of OE is (0-4)/(6-(-2)) = -4/8 = -1/2. The slope of JY is (6-2)/(4-2) = 4/2 = 2/1. The slopes are inverse reciprocals, and (-1/2)(2/1) = -1, so OE is perpendicular to JY.

Therefore JY is the perpendicular bisector of OE.

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories.
Order the softcover or ebook at Amazon.

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!


Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Tuesday, July 02, 2024

June 2024 Geometry Regents Part III


This exam was adminstered in June 2024 .

June 2024 Geometry, Part III

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

32. A building is composed of a rectangular pyramid on top of a rectangular prism, as shown in the diagram below. The rectangular prism has a length of 38 feet, a width of 15 feet, and a height of 22 feet. The rectangular pyramid sits directly on top of the rectangular prism, and its height is 12 feet.

An air purification filter was installed that will clean all the air in the building at a rate of 2400 cubic feet per minute. Determine and state how long it will take, to the nearest tenth of a minute, for the filter to clean the air contained in the building.

Answer:


First you need to find the volume of the building. The building is a pyramid sitting on top of a prism. You have to find the volumes for each of these separately and add them together.

The volume of a rectangular prism is V = (L)(W)(H), and the volume of a rectangular pyramid is V = 1/3(L)(W)(H). Note that the length and width of the two objects are the same but the height are different.

V(prism) = (38)(15)(22) = 12,540 and v(pyramid) = 1/3(38)(15)(12) = 2280. Total volume = 14820.

Next, divide this about by 2400 ft3/min. 14820/2400 = 6.175

To the nearest tenth of a minute, it will take 6.2 minutes.



33. Given: △ABC, △DEF, AB ⟂ BC, DE ⟂ EF, AE ≅ DB, and AC || FD.


Prove: △ABC ≅ △DEF

Answer:


They are expecting a two-column, or a paragraph, proof. Even as a paragraph, all the statements have to be made and the dots connected (so to speak). ou can't just BS a one- or two-sentence rationale which doesn't prove a thing.

Yes, I see a lot of those for every exam.

As always, start with the Given information. Your last statement should be the thing that you are trying to prove, along with a final reason.

Notice that there are two parallel lines, so you will likely have alternate interior angles. There are perpendicular lines, which create right angles, and all right angles are congruent. That means you only need to show that one other pair of sides are congruent to use either ASA or AAS. Since you're given AE = BD, you can pretty much figure out that you'll need to show that AB = DE.

1. △ABC, △DEF, AB ⟂ BC, DE ⟂ EF, AE ≅ DB, and AC || FD. Given.
2. ∠ABC and ∠DEF are right angles. Def of perpendicular lines
3. ∠ABC ≅ ∠DEF All right angles are congruent.
4. ∠A ≅ ∠D Alternate interior angles
5. BE ≅ BE Reflexive property
6. AB ≅ DE Addition postulate of congruence
7. △ABC ≅ △DEF ASA


34. In the diagram below, a boat at point A is traveling toward the most powerful waterfall in North America, the Horseshoe Falls. The Horseshoe Falls has a vertical drop of 188 feet. The angle of elevation from point A to the top of the waterfall is 15°.

After the boat travels toward the falls, the angle of elevation at point B to the top of the waterfall is 23°. Determine and state, to the nearest foot, the distance the boat traveled from point A to point B.

Answer:


Call the base of the waterfall C. To find the length of AB, you have to find the lengths of AC and BC and then subtract AC - BC. You can find both of those lengths using tangent. (Make sure your calculator is in DEGREE mode, or you will get crazy answers.)

Tan 18 = 188 / x

x = 188 / tan 18 = 701.625...



Tan 23 = 188 / y

y = 188 / tan 23 = 442.900...

701.625 - 442.900 = 258.725, or about 259 feet to the nearest foot.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: Portrait of a Lady Vampire & Other Vampiric Cravings, written by Christopher J. Burke, which contains the aforementioned story and three other stories.
Order the softcover or ebook at Amazon.

And don't forget that Burke's Lore, Briefs:A Heavenly Date / My Damned Best Friend is still available!


Also, check out Devilish & Divine, an anthology filled with stories of angels and devils by 13 different authors, and In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.