Geometry Problems of the Day (Geometry Regents, June 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2011

Part IV: A correct answer will receive 6 credits. Partial credit is available.


38. In the diagram below, PA and PB are tangent to circle O, OA and OB are radii, and OP intersects the circle at C.

Prove: ∠AOP ≅ ∠BOP

Answer:

You can prove the two angles are congruent by proving that the triangles are congruent. You know that the tangents make right angles, that the radii are congruent and that they share their hypotenuse. So by the HL Theorem, the two triangles are congruent.

You could also make use of the fact that the two tangents must be congruent, so you could use either SSS or SAS with the tangents, right angles, and radii.

Statement	Reason
1. PA and PB are tangent to circle O, OA and OB are radii, and OP intersects the circle at C	Given
2. ∠OAP is a right angle	Tangents intersect radii at right angles
3. ∠OBP is a right angle	Tangents intersect radii at right angles
4. ∠OAP ≅ ∠OBP	All right angles are congruent
5. OA ≅ OB	All radii in a circle are congruent
6. OP ≅ OP	Reflexive Property
7. Triangle OAP ≅ Triangle OBP	HL Theorem
8. ∠AOP ≅ ∠BOP	CPCTC

End of Part Exam.

More to come. Comments and questions welcome.

More Regents problems.

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