Geometry Problems of the Day (Geometry Regents, August 2024 Part IV)

This exam was adminstered in August 2024 .

August 2024 Geometry, Part IV

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

35. In quadrilateral HOPE below, EH ≅ OP, EP ≅ OH, EJ ≅ OG, and TG and YJ are perpendicular to diagonal EO at points G and J, respectively.

Prove that TG ≅ YJ.

Answer:

This is a long proof. Look at what you are given and what you need to find.

To show that TG = YJ, you need to show that triangle GET ≅ triangle JOY. To show that, you need to use one of the triangle postulates or theorems.

You are given that there are perpendicular lines, which create right angles, and we can show that there's a parallelogram so there's another angle. That means we could do with with either ASA or AAS.

Statement	Reason
1. EH ≅ OP, EP ≅ OH, EJ ≅ OG, and TG and YJ are perpendicular to diagonal EO at points G and J, respectively.	1. Given
2. HOPE is a parallelogram	2. A quadrilateral with two pairs of congruent sides is a parallelogram.
3. Angle JOY ≅ angle GOT	3. Alternate interior angles
4. Angle EGT and angle OJY are right angles.	4. Definition of perpendicular lines
5. GJ ≅ GJ	5. Reflexive Property
6. GE ≅ JO	6. Subtration Postulate
7. Triangle GET ≅ Triangle JOY	7. ASA Postulate
8. TG ≅ YJ.	8. CPCTC

End of Part Exam

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Questions, comments and corrections welcome.

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Geometry Problems of the Day (Geometry Regents, August 2024 Part III)

This exam was adminstered in August 2024 .

August 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 drone is used to measure the size of a brush fire on the ground. Segment AB represents the width of the fire, as shown below. The drone calculates the distance to point B to be 1076 feet at an angle of depression of 25°. At the same point, the drone calculates the distance to point A to be 774 feet at an angle of depression of 36°.

Determine and state the width of the fire, AB, to the nearest foot.

Answer:

Call the right angle C. To find the length of AB, you need to find the lengths of AC and BC and then subtract.

Because the two horizontal lines shown in the picture are parallel (all horizontal lines have slope = 0 and are therefore parallel), then we know that Angle A has a measure of 36° and Angle B has a measure of 25°.

In both cases, we have the hypotenuse and we are looking for the adjacent side of the right triangle. (The opposite side is the height. We don't know it, and we are not looking for it.) This means we must use cosine. Twice.

Cos 25 = x / 1076
x = 1076 cos 25 = 975.1871...

Cos 36 = x / 774
x = 774 cos 36 = 626.1791...

975.1871 - 626.1791 = 349.008

The length of AB is 349 feet.

33. 3 Quadrilateral ABCD has vertices with coordinates A(-3,6), B(6,3), C(6,-2), and D(-6,2). Joe defines an isosceles trapezoid as a trapezoid with congruent diagonals. Use Joe’s definition to prove ABCD is an isosceles trapezoid. [The use of the set of axes below is optional.]

Answer:

You don't have to graph the figure but it will make the problem easier to visualize. Also, you must show that this is an isosceles trapezoid using Joe's method or you will not receive full credit.

First, show that the figure is a trapezoid by showing that there is only one pair of parallel sides with the same slope. The slope of AB is -1, and the slope of CD is -1, so AB || CD. The slope of BC is undefined (a vertical line), and the slope of DA is 4/3, so BC is NOT parallel to DA. Therefore, ABCD is a trapezoid.

Second, in a trapezoid, the diagonals are congruent. so use the Distance Formula (or Pythagorean Theorem) to find the lengths of diagonals AC and BD.

AC = &sqrt;( (-3 - 6)² + (6 - -2)²) = &sqrt;(81 + 64) = &sqrt;(145)

BD = &sqrt;( (-6 - 6)² + (3 - 2)²) = &sqrt;(144 + 1) = &sqrt;(145)

Since AC ≅ BD the trapezoid is isosceles.

If you found that AD is congruent to BC, then you would not get full credit for this problem.

34. Ali made six solid spherical decorations out of modeling clay. Each decoration has a radius of 2.5 inches. The weight of clay is 68 pounds per cubic foot.

Determine and state, to the nearest pound, the total weight of the six decorations.

Answer:

You are given the weight per cubic foot. You need to find the number of cubic feet total in all six decorations. However, you are given measurements in inches, so you'll need to divide by 12³.

V = 4/3 π r³ = 4/3 π (2.5)³ = 65.4498 ...

Six decorations have a volume of 6 * 65.4498 = 392.6990 cubic inches or 392.6990/12³ = 0.227 cubic feet

The weight is 0.227 * 68 = 15.436 or 15 pounds.

End of Part III

How did you do?

Questions, comments and corrections welcome.

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

Geometry Problems of the Day (Geometry Regents, August 2024 Part II)

This exam was adminstered in August 2024 .

August 2024 Geometry, Part II

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

25. In the diagram below, △SUV ~ △TRE.

If SU = 5, UV = 7, TR = 14, and TE = 21, determine and state the length of SV.

Answer:

Because the triangles are similar, then the corresponding sides will be proportional. If we compare the corresponding sides TR and SU, then we will have the scale factor for the dilation from triangle SUV to TRE.

TR/SU = 14/5, which is the scale factor.

Therefore (14/5) SV = 21, which is the length of TE.

Using inverse operations, SV = 21 (5/14) = 15/2 = 7.5

SV = 7.5

You could have created this proportion and solved:

14 / 5 = 21 / x

This would be solved using the same operations.

26. Using a compass and straightedge, construct the line of reflection that maps △ABC onto its image, △DEF. [Leave all construction marks.]

Answer:

The line of reflection will be the perpendicular bisector of the line segment drawn from CF.

Using a straightedge to draw CF. Then create a perpendicular bisector, which is two arcs and a straight line.

First, draw the blue line connecting points C and F.

Second, make a circle centered around C and a circle the same size centered on F. They need to be big enough to overlap in two places. Note: my circles overlap, but they should've been a little bigger to give me more room to work with.

Finally, draw the line of reflection through the two points of intersection in the two circles.

27. Triangle MAX has vertices with coordinates M(-5,-2), A(1,4), and X(4,1). Determine and state the area of △MAX. [The use of the set of axes below is optional.]

Answer:

Plotting the points and drawing the triangle will not get you any credit, but it will make it easier to visualize the problem.

Once you have the triangle plotted, draw a rectangle around the triangle so that M is a corner of the rectangle and A and X are on the sides of the rectnagle. You know have four rectangles.

You can figure out the size of the three outer triangles using 1/2 * base * height and counting the boxes. Find the area of the rectangle, and subtract the area of the three outer triangles. Whatever is left is the area of MAX.

The area of the rectangle is Length * Width = 9 * 6 = 54.

The area of the three outer triangles are 1/2(6)(6) = 18, 1/2(3)(3) = 4.5, and 1/2(9)(3) = 13.5.

Therefore, the area of MAX = 54 - 18 - 4.5 - 13.5 = 18.

You could find the base and the height of the triangle. Note: this is a two-point question, so the answer shouldn't be complicated.

To find the any of a triangle, you need to find the length of the base and the length of the altitude (or height). Those two lines need to be perpendicular to each other. You can use the distance formula to find these.

As soon as the image is graphed, we can see that MA has a slope of 1, it has a rise of 1 and a run of 1, and 1/1 = 1. We can also see that AX has a slope of -1 because it was a rise of -1 and a run of 1, and -1/1 = -1. Since (1)(-1) = -1, then MA is perpedicular to AX, so their lengths can be used as the base and the height.

Use the distance formula or the Pythagorean Theorem to find the lengths of these segments.

MA = &sqrt;(6² + 6²) = &sqrt;(72). AX = &sqrt;(3² + 3²) = &sqrt;(18).

The area of the triangle is 1/2 * base * altitude = 1/2 &sqrt;(72) &sqrt;(18) = 1/2 &sqrt;(1296) = 1/2 (36) = 18.

28. A person observes a kite at an angle of elevation of 32° from a line of sight that begins 4 feet above the ground, as modeled in the diagram below. At the moment of observation, the kite is 70 feet above the ground.

Determine and state the horizontal distance, x, between the person and the point on the ground directly below the kite, to the nearest foot.

Answer:

You have the opposite side and you are looking for the adjacent side, so you need to use the TANGENT ratio. The Trick to this question is that you have to remember to subtract 4 from the 70 feet above the ground because the kite is only 66 feet above the observer.

Tan 32 = 66 / x
x = 66 / (Tan 32) = 105.622...

The distance along the ground is 106 feet, to the nearest foot.

29. In △AGL below, N and E are the midpoints of AG and AL, respectively, NE is drawn.

If NE = 15 and GL = 3x - 12, determine and state the value of x.

Answer:
The midsegment (the line connecting two midpoints) is equal to half of the length of the third side of the triangle. You can write an equation that says that double the size of NE is equal to the length of GL.

2(15) = 3x - 12
3x - 12 = 30
3x = 42
x = 14.

30. In the diagram below, △TAN is the image of △SUN after a reflection over NZ.

Use the properties of rigid motions to explain why △TAN ≅ △SUN.

Answer:
In a rigid transformation, the image is the same size and shape as the pre-image. Since distance is preserved the corresponding sides must be congruent. Therefore by SSS, the triangles must be congruent.

Note that in a question like this, I'm never sure exactly how much of an explanation that they are looking for. They basically want a definition of rigid transformation but it has to address this problem specifically.

31. A pyramid with a square base is made of solid glass. The pyramid has a base with a side length of 5.7 cm and a height of 7 cm. The density of the glass is 2.4 grams per cubic centimeter.

Determine and state, to the nearest gram, the mass of the pyramid.

Answer: You need to find the Volume of the pyramid and then using the density and the Volume, find the mass of the pyramid.

The Volume of a pyramid is 1/3 * the Area of the base * the height. The base is a square, so it's area is (5.7)², or 32.49. Therefore the Volume is 1/3 * (32.49) * (7) = 75.81.

Density is equal to mass divided by the Volume (d = m/V), so the mass is equal to Density times Volume (m = Vd). Therefore, the mass = (75.81) * (2.4) = 181.944, which is 182 grams to the nearest gram.

End of Part II

How did you do?

Questions, comments and corrections welcome.

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

Algebra Problems of the Day (Algebra Regents, August 2024 Part IV)

This exam was adminstered in August 2024 .

August 2024 Algebra, Part IV

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

35. Jen joined the Fan Favorite Movie Club at the local movie theater. At this theater, the cost of admission in May and June remains the same. In May, she saw 2 matinees and 3 regular-priced shows and spent $38.50. In June, she went to 6 matinees and one regular-priced show and spent $47.50.

Write a system of equations to represent the cost, m, of a matinee ticket and the cost, r, of a regular-priced ticket.

Jen said she spent $5.75 on each matinee and $9 on each regular show. Is Jen correct? Justify your answer.

Use your system of equations to algebraically determine both the actual cost of each matinee ticket and the actual cost of each regular ticket.

Answer:

You need to write a system of equations that models the prices of the movie tickets. Substitute her numbers into the equations to see if they work. (They probably won't because of the next part of the problem.) Then solve that system using elimation or substituion.

The system of equations that models the ticket buying is as follows:

2m + 3r = 38.50
6m + 1r = 47.50

Jen believes that m = 5.75 and r = 9, so plug those values in:

2(5.75) + 3(9) = 38.50
6(5.75) + 1(9) = 47.50

38.50 = 38.50 (check)
43.50 =/= 47.50 (does not check)

Jen is incorrect because her values don't work for the month of June.

You can use substitution with the second equation, or you could multiply either equation by 3 and subtract.

2m + 3r = 38.50
6m + 1r = 47.50

6m + 9r = 115.50
6m + 1r = 47.50
8r = 68.00
r = 8.50

2m + 3(8.50) = 38.50
2m + 25.50 = 38.50
2m = 13.00
m = 6.50

Check: 6(6.50) + 8.50 = 47.50 (Check)

A matinee ticket is $6.50 and a regular ticket is $8.50.

Using substitution:

2m + 3r = 38.50
6m + 1r = 47.50
r = -6m + 47.50

2m + 3(-6m + 47.50) = 38.50

2m - 18m + 142.50 = 38.50
-16m = -104.00
m = 6.50

2(6.50) + 3r = 38.50
13 + 3r = 38.50
3r = 25.50
r = 8.50

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

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

Algebra Problems of the Day (Algebra Regents, August 2024 Part III)

This exam was adminstered in August 2024 .

August 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. The owner of an ice cream stand kept track of the number of ice cream cones that were sold each day of the first week in June. She compared the ice cream sales to the average daily temperature. The data are shown in the table below.

State the linear regression equation for these data, rounding all values to the nearest hundredth.

State the correlation coefficient, to the nearest hundredth, for the line of best fit for these data.

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

Answer:

You have to perform a Linear Regression in your calculation. Put the data from the table into two lists (L1 and L2).

After rounding the values in your calculator, you will get the equation y = 15.13x - 959.63.

The correlation coefficient, r, is 0.99.

This correlation coefficient indicates that there is a strong postivie correlation in the data.

32. Graph the system of inequalities on the set of axes below:

y > 3x - 4
x + 2y <6
Label the solution set S.
Is the point (2,2) a solution to the system? Justify your answer.

Answer:

Remember that > means a broken line with shading about the line and that < means a solid line with shading below that line.

You can use your calculator to graph the lines and use the Table of Values. Remember that the first line must be broken or dotted.

Rewrite the second equation as y < -1/2 x + 3.

Your graph will look like this:

The point (2,2) is NOT is the solution set of the system of inequalities because it is on the broken line. The points on the broken line are NOT part of the solution.

Note: your final answer will depend upon the graph you draw. If you have a graphing error, then the answer to the final part of the question MUST match your graph.

33. An object is launched upward at 64 feet per second from a platform 80 feet above the ground. The function s(t) models the height of the object t seconds after launch.

If s(t) = -16t² + 64t + 80, state the vertex of s(t), and explain in detail what each coordinate means in the context of the problem.

After the object is launched, how many seconds does it take for the object to hit the ground? Justify your answer.

Answer:

This is a problem of gravity, which creates an upside-down parabola when graphed. The vertex is a point on the Axis of Symmetry, which can be found using x = -b/(2a).

The Axis of Symmetry is x = -64/(2(-16)) = 2.

Calculate s(2) = -16(2)² + 64(2) + 80 = 144. The coordinates of the vertex are (2,144).

In the context of this problem, that point says that at two seconds the object reaches its maximum height, which is 144 feet above the ground.

The object will hit the ground at when -16t² + 64t + 80 = 0.

-16t² + 64t + 80 = 0
t² - 4t - 5 = 0
(t - 5)(t + 1) = 0
t - 5 = 0 or t + 1 = 0
t = 5 or t = -1

Discard the negative answer because time cannot be negative. The object hits the ground at t = 5 seconds.

34. Solve the system of equations algebraically for all values of x and y.

y = x² + 4x - 1
y = 2x + 7

Answer:

Set the two expressions equal to each other. Solve the quadratic equation. Then solve for y.

x² + 4x - 1 = 2x + 7
x² + 2x - 8 = 0
(x + 4)(x - 2) = 0
x + 4 = 0 or x - 2 = 0
x = -4 or x = 2

If x = -4, y = (-4)² + 4(-4) - 1 = -1.

If x = 2, y = (2)² + 4(2) - 1 = 11.

End of Part III

How did you do?

Questions, comments and corrections welcome.

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

Algebra Problems of the Day (Algebra Regents, August 2024 Part II)

This exam was adminstered in August 2024 .

August 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. On the set of axes below, graph f(x) = x² + 4x + 1.
State the coordinates of the minimum.

Answer:

You can use your graphing calculator and check the Table of Values. Be sure to state the coordinates of the minimum. Label it if you have any other points listed.

The minimum point is (-2, -3).

26. If f(x) = 30x²/(x + 2), determine f(1/2).

Answer:

Substitute and evaluate. Remember the order of operations. Or put it into your calculator. (If anyone is reading all of my Algebra posts, you might have noticed that I wrote the exact same thing for Question 26 on the June 2024 Regents becuase they asked a Very Similar Question!)

f(1/2) = 30 (1/2)² / (1/2 + 2) = (30)(1/4) / 2.5 = 3

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 = 30x²/(x + 2) and check the table of values for x = .5.

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. Explain why the relation shown in the table below is a function.

Complete the table below with values for both x and y so that this new relation is not a function.

Answer:

The relation is a function as long as the x values do not repeat. (Again! I wrote this very same comment in June for Question 27!) Since the x values don't repeat, it is a function.

In the second table, repeat any of the values of the first line in the first box. In the second box, write a different number. In other words, 0 and 6 is a good answer, but 0 and 4 is not because (0,4) is already in the relation. 2 and 4 is a good answer, but 2 and 5 is not.

28. Solve algebraically for x: 0.05(x - 3) = 0.35x - 7.5

Answer:

Multiply the equation by 20 to get rid of the 0.05. (Or divide everything by 0.05, which is the same thing.) Then move the variables to one sides and the numbers to the other.

0.05(x - 3) = 0.35x - 7.5
x - 3 = 7x - 150
147 = 6x
24.5 = x

29. Use the quadratic formula to determine the exact roots of the equation x² + 3x - 6 = 0.

Answer:
If it says to use the quadratic formula, then you must use it. If you use any other method, then the most you can get is 1 point for a correct value of x.

The quadratic formula is x = (-b + √(b² - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation. In this case, a = 1, b = 3, and c = -6.

x = (-b + √(b² - 4ac)) / (2a)
x = (-3 + √(3² - 4(-6))) / (2)
x = (-3 + √(9 + 24)) / (2)
x = -3/2 + √(33) / 2

Those are the exact values. Do not approximate by changing them to decimals.

30. Factor 5x³ - 80x completely.

Answer:
Whenever you see "Factor completely", you can be pretty sure that there will be more than one step. Show them all. Once again, I copied text from Question 30 of the June Regents!)

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.

5x³ - 80x = 5x (x² - 16) = 5x (x + 4)(x - 4)

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