(x, why?): Geometry Problems of the Day (Geometry Regents, June 2025 Part IV)

Friday, July 04, 2025

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

This exam was adminstered in June 2025.

June 2025 Geometry, Part IV

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

35. Given: Quadrilateral CARI with CA ≅ RI and CI ≅ RA, and AEI and LEH are drawn.

Prove: HA * EL = LI * EH

Answer:

You need to write a proof that shows, first of all, that CARI is a parallelogram. You cannot assume that it is. Then you are going to show that there are two triangles that are similar, so their corresponding sides are proportional. THEN you are going to cross-multiply BUT YOU ARE NOT GOING TO SAY "cross multiply" because that is a NO-NO. (I'm not kidding. See below.)

Statement	Reason
1. CA ≅ RI and CI ≅ RA	Given.
2. CARI is a parallelogram.	Opposite sides of the quadrilateral are congruent.
3. ∠HAE ≅ ∠LIE	Alternate interior angles of parallel lines are congruents
4. ∠HEA ≅ ∠LEI	Alternate interior angles of parallel lines are congruents
5. △HEA ~ △LEI	AA Theorem
6. HA / EH = LI / EL	Corresponding sides of similar traingles are proportional.
7. HA * EL = LI * EH	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.

End of Part IV

End of Exam

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