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

This exam was adminstered in January 2025.

January 2025 Geometry, Part III

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

32. Modeled by right triangles below, a surveyor (S) is taking land measurements using a cabin (C), a boulder (B), and a tree (T) as fixed points of reference. The cabin, boulder, and tree are collinear. The surveyor is 917 meters from the cabin, 1048 meters from the boulder, and 1425 meters from the tree.

Determine and state, to the nearest degree, the measure of ∠BST.

Answer:

CST is a right triangle. CSB is also a right triangle. BST is not a right triangle, so you can't you one trig function. However, you can use TWO trig functions. If you find the measure of ∠CST and then subtract the measure of ∠CSB, the difference will be the measure of ∠BST. In each case, we have the adjacent side (917) and the hypotenuse (1425 and 1048), so we need to use cosine two times.

cosine x = 917/1425
x = cos^-1(917/1425) = 49.946 degrees

cosine y = 917/1048
y = cos^-1(917/1048) = 28.955 degrees

Subtract 49.946 - 28.955 = 20.991. So the measure of ∠BST is 21 degrees to the nearest degree.

33. A garden bed, pictured below, is a square prism with a rectangular prism taken out. The inside length of the square prism is 6 feet. The rectangular prism taken out has a width of 2 feet and a length of 4 feet.

The diagram below shows the top view of the garden bed with its inside measurements.

The garden bed is filled with topsoil to a uniform height of 1.25 feet.

Determine and state the volume of the topsoil, in cubic feet.

Each bag of topsoil sells for $3.68 and contains 2 cubic feet of topsoil.

Determine and state the total cost of the bags of topsoil that must be purchased to fill the garden.

Answer:

TO find the area of the U-shaped garden bed, either break it into three rectangles and find each area separately, OR draw a line on the bottom to create a square, find the area of the square and subtract the rectangle in the middle.

Whichever way you do it, the area of the base is 28 square feet.

The Volume is the Area of the Base times the height, so 28 * 1.25 = 35 ft³.

If each bag of topsoil contains 2 cubic feet of topsoil, then you need 35/2 = 17.5 or 18 bags. You need to ROUND UP to the next integer. You have to buy a Whole Number of bags and if you round down, you will not have enough topsoil to finish the job.

If you buy 18 bags at $3.68 each then (18)(3.68) = $66.24

34. Given: △ACD with ABC, AED, and BE || CD

Prove: AB • AD = AE • AC

Answer:

This seems like an odd thing to prove. Why do you care if these products are equal? But if you divide both sides of this equation by AC and AD, we get a proportion of corresponding sides of the triangles. So we need to first show that these triangles are similar, so that we can prove something about the cross products.

Statement	Reason
1.BE || CD	Given
2. ∠ABE ≅ ∠ACD	Alternate interior angles of parallel lines are congruents
3. ∠AEB ≅ ∠ADC	Alternate interior angles of parallel lines are congruents
4. △ABE ~ △ACD	AA Theorem
5.AB / AC = AE / AD	Corresponding sides of similar traingles are proportional.
6. AB • AD = AE • AC	The product of the means equals the product of the extremes.

You could also have stated that "Cross products are equal." However, "cross multiply" was NOT acceptable. The best that I could figure is that the former is a reason while the latter is a command.

Also, you could have used that ∠A ≅ ∠A by the Reflexive Property in place of Step 2 or Step 3.

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 other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Geometry Problems of the Day (Geometry Regents, June 2025 Part II)

This exam was adminstered in June 2025.

June 2025 Geometry, Part II

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

25. In the year 2020, the village of Depew, New York had an area of 5.1 square miles and a population of 15,069. In the same year, the village of Lancaster, New York had an area 2.7 square miles and a population of 10,087.

Which village had the larger populatio density in 2020? Justify your answer.

Answer:

Density is mass divided by volume, but population density is population divided by square footage. Find the density of both cities.

The population density of Depew is 15069/5.1 or about 2954.7.

The population density of Lancaster is 10087/2.7 or about 3735.9.

Lancaster had the larger population density.

You only got both credits if you answered the question and showed work. An answer of "Lancaster" without any work is woth 0 points.

26. In △ABC below, AC is extended through C to D, m∠A = (3x - 22)^o, m∠A = (4x - 18)^o, and m∠BCD = (6x - 23)^o.

Determine and state m∠ACB.

Answer:

This is a two-part problem and, if you ask me (and no one did), a little too involved for a 2-point question.

The exterior angle is equal to the sum of the two remote angles. You need to write an equation to solve for x. Then you have to find the size of angle BCD. AND THEN you have to find the measure of the angle supplementary to the exterior angle.

Again, a little much.

3x - 22 + 4x - 18 = 6x - 23
7x - 40 = 6x - 23
x = 17

This means that the exterior angle is 6(17) - 23 = 79. So angle ACB = 180 - 79 = 101 degrees.

27. Parallelogram ABCD is shown below. Using a compass and straightedge, construct the altitude from point A to side DC. [Leave all construction marks.]

Answer:

From point A, draw an arc the length of AD. OR extend line CD and draw an arc of any length that intersects CD in two places.

From the two points of line CD, make arcs below CD. Make a point where the two arcs intersect. Call it anything, say E. with a straightedge, draw AE.

That's it.

If you made the altitude from points B, C, or D, you only got one point. If you did it from any other point, or if you made a perpendicular bisector of CD, you score no points.

28. Quadrilateral QUAD is graphed on the set of axes below.

Determine and state the area of quadrilateral QUAD.

Answer:

It was quite easy to show that the diagonals of the quadrilater were perpendicular to each other because the diagonals were vertical and horizontal lines. However, the question did not say that this was a parallelogram. It only said it was a quadrilateral.

It is also easy to see that QUAD could be broken up into four congruent triangles, for which the area could be found.

HOWEVER, this past January, there was a similar question where the figure was NOT so nice. As a result, the easiest way to find the area was to draw a rectangle box around the figure, find the area of individual triangles between the shape and the rectangle, and then subtract. What remained was the area of the figure. That wasn't necessary to do her, BUT anyone who practiced that exam in the days before this exam would've been more likely to take this approach.

I will say that I saw a few answers based on the fact that it is a rhombus. To use those formulas, however, you needed to show that it was a rhombus. This was a lot of extra work for 2 points, but more power to whoever did it this way!

If you draw the two diagonals, you will split QUAD into four triangles. Each of those triangles has a base of 5 and a height of 3.

So Area = 4 (1/2) (3) (5) = 30

If you found the area of the rectangle around QUAD, it had a length of 10 and a width of 6, for an Area of 60. If you then subtracted the four triangles, each of which was 5 by 3, you got 60 - 30 = 30.

Common mistakes were finding the perimter, or finding the lengths of the sides and then sqauring it, assuming that the shape was a square.

29. In a right triangle, the acute angles have the relationship sin(3x - 7)^o = cos(x + 1)^o

Determine and state the value of x.

Answer:
If the sine value is equalt to the cosine value than the sum of the two angles must be 90 degrees. (The "co" is "cosine" means "complementary", and complementray angles add up to 90 degrees.

3x + 7 + x + 1 = 90
4x - 6 = 90
4x = 96
x = 24

Common mistakes are to set the two expressions equal to each other, which is a conceptual error, setting the sum equal to 180, another conceptual error, or any computational mistakes solving the algebraic equation.

One mistake costs one point. Two points results in a score of zero.

30. In circle A below, m∠BAM = 36°

If AB = 20, determine and state the length of arc MB. [Leave your answer in terms of π.]

Answer:
The circumference of a full circle is 2πr. For an arc, it is the circumference times the fraction of the circle, which is denoted by the measure of the circle arc divided by 360 degrees.

So 36/360 (2)(20)π = 4π.

That's all there was to this.

31. In triagnels ANT and ELM below, AN = 6, NT = 5.6, TA = 4, EL = 9, LM = 8.4, and ME = 6.

Explain why △ANT ~ △ELM

Answer: You can show that two triangles are similar by showing that all three pairs of corresponding sides are proportional by SSS. Also, one triangle is the image of the other after a dilation of a particular scale factor. ("Obviously", saying this is not enough. You need to provide the numbers.)

6/4 = 1.5, 9/6 = 1.5, and 8.4/5.6 = 1.5.

Triangle ELM is a dilation of ANT by a scale faction of 1.5. All of the corresponding sides are proportional so the two are similar by SSS.

Note: if SSS wasn't mentioned, you likely lost 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 other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Geometry Problems of the Day (Geometry Regents, January 2025 Part II)

This exam was adminstered in January 2025.

January 2025 Geometry, Part II

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

25. In the diagram below, quadrilateral BCDE maps onto quadrilateral JKLM using a sequence of rigid motions.

Determine and state the degree measure of angle D.

Answer:

Angle B is congruent to Angle J. Angle C is congruent to Angle K. The sum of the four angles in a quadrilateral is 360 degrees.

70 + 117 + 91 = 278 and 360 - 278 = 82 degrees, so the measure of angle D is 82 degrees.

26. Given AB below, use a compass and a straightedge to construct a segment that is 1/4 AB. [Leave all construction marks.]

Answer:

Construct a perpendicular bisector to find the midpoint. Then find construct another perpendicular bisector between A and the midpoint. Half of a half is a quarter.

If you label the new point C, then AC = 1/4 AB.

The following illustration was provided by New York State.

27. A dog sees a bird in a tree. The angle of elevation from the dog’s eyes to the bird is 36°, as modeled below.

The dog is 18.5 feet away from the base of the tree, and his eyes are 2.5 feet above the ground.
Determine and state how high the bird is above the ground, to the nearest foot.

Answer:

In the right triangle shown, you have the base, which is adjacent to the angle, and you want to find the height, which is the opposite of the angle. This means that you want to use TAN to find the height. And then you have to add 2.5 feet because the triangle isn't on the ground.

tan 36 = x/18.5
x = 18.5 tan 36 = 13.44..

Add 13.4 + 2.5 = 15.9, which is 16 feet to the nearest foot.

28. Pure silver has a density of 10.5 g/cm³. Samantha has a pure silver charm on her necklace in the shape of a sphere. The radius of the charm is 0.5 cm.
Determine and state the mass of the charm, to the nearest tenth of a gram.

Answer:

You have the density. You need to find the Volume so you can find the mass, using the formula D = m/V.

V = 4/3 π r³ = 4/3 (0.5)³ = 0.523...

Use this Volume in the Density formula to find the Mass.

D = m/V
10.5 = m / 0.523
m = (10.5)(0.523) = 5.4915.

The mass is 5.5 grams.

29. In △ABC, below, DE is drawn such that AD = 4, DB = 8, AE = 3, and EC = 6.

Explain why △ADE ~ △ABC.

Answer:
You may recall that a midsegment connects the midpoints of two sides of a triangle and is parallel to the third side. The property of being parallel to the third side is true for any segment that connects two points that have the same proportionality as well.

3/6 = 1/2, so AE:EC = 1:2, and 4/8 = 1/2, so AD:DB = 1:2.

Since ED connects two points with the same proportionality, ED || CB. Since the lines are parallel then ∠C ≅ ∠AED and ∠D ≅ ∠ADE. (Also, ∠A is congruent to itself.) By AA, △ADE ~ △ABC.

30. In circle E below, tangent PA and secant PBC are drawn.

If PB =9 and BC = 16, determine and state the length of PA.

Answer:
The Tangent-Secant Theorem tells us that the square of the tangent is equal to the product of the length of the secant line times the portion of the line that is outside the circle. (If you think about it, you are also multiplying the length of the tangent by the portion that is outside the circle!)

(PA)² = (PB)(PC) = (9)(9 + 16) = (9)(25) = (225)
PA = &sqrt;(225) = 15

Note that the square root of 9 is 3 and the square root of 25 is 5, and that 3 * 5 = 15. That works as well.

31. In a right triangle, sin(4x + 3)° = cos(2x - 9)°. Determine and state the value of x.

Answer: If the sine of one angle is equalt to the cosine of another angle, then the sum of the two angles is 90 degrees. The "co" in "cosine" stands for "complementary", and complementary angles add up to 90 degrees.

4x + 3 + 2x - 9 = 90
6x - 6 = 90
6x = 96
x = 16

End of Part II

How did you do?

Questions, comments and corrections welcome.

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