August 2022 Geometry Regents, Part IV

The following are questions from the recent August 2020 New York State Common Core Geometry Regents exam.

August 2022 Geometry, Part IV

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

35. Given: Quadrilateral ABCD, AC and EF intersect at H, EF || AD, EF || BC, and AD ≅ BC.

Prove: (EH)(CH) = (FH)(AH)

Answer:
This is an unusual proof, in my opinion. Usually, the final line is some theory of congruence, like SAS or CPCTC, etc. This one shows two products. However, it's really the cross-products of a proportion. Looking at the figure, the proportion seems to be created by the corresponding sides of similar triangles. Once you have that proportion, the final line comes from the product of the means and the extremes being equal.

Statement	Reason
1. Quadrilateral ABCD, AC and EF intersect at H, EF || AD, EF || BC, and AD ≅ BC.	Given
2. AD || BC	If two lines are parallel to a given line then they are parallel to each other. (Transitive property)
3. ABCD is a parallelogram	If the opposite sides of a quadrilateral are parallel then the quadrilateral is a parallelogram.
4. AB || CD	Opposite sides of a parallelogram are parallel.
5. ∠BAC ≅ ∠DCA	Alternate interior angles are congruent.
6. ∠AHE ≅ ∠CHF	Vertical angles are congrunt
7. △AHE ~ △CHF	AA Postulate
8. EH/FH = AH/CH	If two triangles are similar, then their corresponding sides are proportional.
9. (EH)(CH) = (FH)(AH)	The product of the means = the product of the extremes.

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

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