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6. In the diagram of circle O, tangent PB is drawn as is secant PAC. We wish to prove the Secant/Tangent Length Theorem, i.e. PA·PC = PB2. (a) Explain why APAB is similar to APBC (i.e. explain which two angles pairs must be congruent and why). O® A P B (6) Set up a proportion involving PA, PB, and PC based on the similar triangles and rewrite it in product form. Draw the two similar triangles separately and in the same orientation to help. COMMON CORE GEOMETRY, UNIT #8 - CIRCLE GEOMETRY - LESSON #8 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2018

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MrRoyal MrRoyal · Answer 1 · 2022-05-09T16:08:31+02:00

The lines PC and PB are the secant and the tangent lines of the circle, while the proportion involving PA, PB, and PC is PA * PC = PB²

The triangles are given as:

Δ PAB and Δ PBC

Both triangles have a common point at P.

This means that:

∠P ≅ ∠P --- reflexive property of congruent angles

Also, the angles at C and B are congruent:

This is represented as:

∠B ≅ ∠C

Both triangles have a common side length PB.

Hence, the triangles are similar by ASA similarity theorem

The similarity theorem in (a) is the ASA similarity theorem.

This means that:

PA : PB = PB : PC

Express as fraction

[tex]\frac{PA}{PB} = \frac{PB}{PC}[/tex]

Cross multiply the expressions

PA * PC = PB * PB

Evaluate the product

PA * PC = PB²

Hence, the proportion involving PA, PB, and PC is PA * PC = PB²

Read more about secant and tangent lines at:

https://brainly.com/question/14962681

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