This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.
January 2023 Geometry, Part III
Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.
32.
Sally and Mary both get ice cream from an ice cream truck. Sally’s ice cream is served as a
cylinder with a diameter of 4 cm and a total height of 8 cm. Mary’s ice cream is served as a
cone with a diameter of 7 cm and a total height of 12.5 cm. Assume that ice cream fills
Sally’s cylinder and Mary’s cone.
Who was served more ice cream, Sally or Mary? Justify your answer.
Determine and state how much more is served in the larger ice cream than the smaller ice cream, to the nearest cubic centimeter.
Answer:
Calculate the volume of the cylinder and the volume of the cone. Compare them and then subtract to get the second answer. Make sure to use the radius and not the diameter.
Volume of cylinder: π (2)2(8) = 100.53
Volume of cone: (1/3) π (3.5)2(12.5) = 160.35
Difference: 160.35 - 100.53 = 59.82
Mary got 60 cc more ice cream.
You must state the name or you will lose a credit for not answering the question.
33. Given: ∆AEB and ∆DFC, ABCD, AE || DF, EB || FC, AC ≅ DB
Prove: ∆EAB ≅ ∆FDC
Answer:
The parallel lines and transversals give you alternate interior and exterior angles. The fact that AC = BD gives you the fact that says AB and CD must be congruent after using the Reflexive Property and the subtraction property.
| Statement | Reason |
| 1. ∆AEB and ∆DFC, ABCD, AE || DF, EB || FC, AC ≅ DB | Given |
| 2. ∠A ≅ ∠D | Alternate Interior Angles |
| 3. ∠ABE ≅ ∠DCF | Alternate Exterior Angles |
| 4. BC ≅ BC | Reflexive Property |
| 5. AB ≅ CD | Subtraction Postulate |
| 6. ∆EAB ≅ ∆FDC | ASA |
34. Barry wants to find the height of a tree that is modeled in the diagram below, where ∠C is a
right angle. The angle of elevation from point A on the ground to the top of the tree, H, is 40°.
Barry claims that ∆ABH is isosceles. Explain why Barry is correct.
Determine and state, to the nearest foot, the height of the tree.
The angle of elevation from point B on the ground to the top of the tree, H, is 80°. The
distance between points A and B is 85 feet.
Answer:
You can't talk and "explain" your way out of this without backing it up with some data, some facts. All that a paragraph of text will get you is the chance to say something incorrect.
Angle HBC is 80 degrees, so Angle ABH is 100 degrees. This means that angle AHB is 40 degrees.
Note that the Exterior Angle Theorem tells us that AHB = 80 - 40, which is 40 degrees.
Since Angle A and Angle AHB are congruent, the triangle is isosceles.
Since ABH is isosceles, then HB is 85 feet. This means we can use the sine ratio to find the height of the tree.
sin 80 = x / 85
x = 85 * sin 80 = 83.7
The tree is 84 feet tall.
End of Part III
How did you do?
Questions, comments and corrections welcome.
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