(x, why?): 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.