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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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Section
BuyFindarrow_forward

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

In Review Exercises 27 to 29, give a proof for each statement.

G i v e n :                 A P -   a n d   B P -   a r e   t a n g e n t   t o   Q   a t   A   a n d   B

                                                              C   i s   t h e   m i d p o i n t   o f   A B ^

            P r o v e :                     P C   b i s e c t s   A P B

To determine

To Prove: PC bisects APB, if AP- and BP- are tangent to Q at A and B. C is the midpoint of AB^.

Explanation

Concept:

Theorem 6.1.3:

In a circle (or in congruent circles), congruent minor arcs have congruent central angles.

Theorem 6.1.3:

In a circle (or in congruent circles), congruent central angles have congruent minor arcs.

Theorem 6.1.5:

In a circle (or in congruent circles), congruent chords have congruent minor (major) arcs.

Theorem 6.1.6:

In a circle (or in congruent circles), congruent arcs have congruent chords.

Theorem 6.1.7:

Chords that are at the same distance from the center of a circle are congruent.

Theorem 6.1.8:

Congruent chords are located at the same distance from the center of a circle.

Theorem 6.2.4:

The measure of an angle formed by a tangent and a chord drawn to the point of tangency is one-half the measure of the intercepted arc.

Theorem 6.3.4:

The tangent segments to a circle from an external point are congruent.

Calculation:

Given AP- and BP- are tangent to Q at A and B. C is the midpoint of AB^.

By Theorem 6.3.4:

The tangent segments to a circle from an external point are congruent.

So, AP=BP

Since, C is the midpoint of AB^.

So, AC^=CB^

By, Theorem 6.1.6:

In a circle (or in congruent circles), congruent arcs have congruent chords.

Because congruent chords, AC=CB.

Theorem 6.1.3:

In a circle (or in congruent circles), congruent minor arcs have congruent central angles.

Since, AC^=CB^ then AQC=BQC

By Theorem 6

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