Geometry Problems of the Day (Geometry Regents, June 2026 Parts III & IV)

This exam was adminstered in June 2026.

June 2026 Geometry, Part III

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

32. As modeled below, Maria wants to determine the height of the building across the street from her position, M. The angle of elevation from M to the top of the building, T, is 36°. From M, the angle of depression to the base of the building, B, is 18°. The buildings are 80 feet apart.

Determine and state, to the nearest foot, the height of the building, TB, across the street from Maria.

Answer:

There are two right triangles. In both cases, you know the adjacent side and you are looking for the opposite side. You need to use the tangent ratio twice (once for each angle) and then add the two heights together.

tan 18 = x/80
x = 80 tan 18 = 25.99

tan 36 = x/80
x = 80 tan 36 = 58.12

TB = 25.99 + 58.12 = 84.11

The height is 84 feet.

33. An artist uses clay to make solid pyramids with a square base whose sides measure 12 cm, as modeled below.

The height of each pyramid is 8 cm, and the density of the clay is 1.25 g/cm³. If the artist has 15 kilograms of clay, determine and state the maximum number of pyramids that can be made.

Answer:

Find the Volume of one pyramid and use that and the density to find the mass of one pyramid. And then divide to find the number that can be made. Be careful with your units.

V = 1/3 (12)(12)(8) = 384.

m = VD = (384)(1.25) = 480 grams = .48 kg

15 / .48 = 31.25

A maximum of 31 pyramids can be made. Remember that you have to round down because there isn't enough clay to round up. (If you did the math correctly, you would've rounded down anyway.)

34. Given: Triangle ABC, AB ≅ AC, DE ⊥ BC, and HF ⊥ BC

Proce DE / BE = HF / CF

Answer:

Write a prove that shows that the two triangles are similar so that the corresponding sides of the similar triangles are proportional.

Statement	Reason
1. Triangle ABC, AB ≅ AC, DE ⊥ BC, and HF ⊥ BC	Given.
2. △ABC is isosceles.	A triangle with two congruent sides is isosceles.
3. ∠B ≅ ∠C	The base angles of an isosceles triangle are congruent.
4. ∠DEB and ∠HFC are right angles	Definition of perpendicular
5.∠DEB ≅ ∠HFC	All right angles are congruent.
6. △DEB ~ △HFC	AA Theorem
7. DE / BE = HF / CF	Corresponding sides of the similar triangles are proportional

June 2026 Geometry, Part IIV

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

35. Triangle ABC has vertices whose coordinates are A(-3, 21), B(-5, 2), and C(-1, 8).

State the coordinates of point D such that quadrilateral ABCD is a parallelogram.
[The use of the set of axes on the next page is optional.]

Prove ABCD is a parallelogram.
[The use of the set of axes on the next page is optional.]

State the coordinates of point E, the midpoint of BC.
[The use of the set of axes on the next page is optional.]

Prove △ABE is an isosceles triangle. [The use of the set of axes below is optional.]

Answer:

If they give you a grid, use it. It counts as work, and the visual is helpful. It's easier to count boxes than to remember and apply formulas.

To show that find point A, look at the change in x and y from B to C. Apply the same changes from A to find D.

From B to C is 4 to the right and 6 units up. If you go 4 to the right and 6 units up from A, you get D(1,5).

You can prove that it's a parallelogram by showing that the opposite slopes are equal which makes the lines parallel.

Slope BC = 6/4 = 3/2
Slope AD = 6/4 = 3/2
Slope AB = 3/-2 = -3/2
Slope CD = -3/2
Two pairs of opposite sides have the same slope and are parallel. Therefore ABCD is a parallelogram.

The midpoint of BC is the halfway point between B and C. We already know that C is 4 to the right and 6 up from B. Point E is half of those distances, 2 to the right and 3 up. That puts the point at E(-3,5)

To prove △ABD is isosceles, you can use the grid instead of a formula to find the lengths of AB and BE. AE is 6.

AB = √(2² + 3²) = √(13)

BE = √(2² + 3²) = √(13)

ABE is an isosceles triangle because 2 sides are congruent.

End of Exam

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