   Chapter 6.1, Problem 19E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

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

Circles O and Q have the common chord A B ¯ . If A B = 6 , ⊙ O has a radius of length 4 , and ⊙ Q has a radius of length 6 , how long is O Q ¯ ? Exercises 18, 19

To determine

To find:

The length of OQ¯.

Explanation

Given:

Circles O and Q have the common chord AB¯. The circle with center O has a radius of length 4 and the circle with center Q has a radius of length 6. The length of AB is 6.

Theorem used:

A radius that is perpendicular to the chord bisects the chord.

An angle inscribed in a semicircle is a right angle.

If X is a point on AB¯ and A-X-B, then AX¯+XB¯=AB¯.

Calculation:

In the circle O, OA¯ and OB¯ are radii and in circle Q, QA¯ and QB¯ are radii, where OA¯=4 and QA¯=6. Thus we get a kite with diagonals OQ¯ and AB¯.

We know that the diagonals of a kite are perpendicular to each other.

Hence OQ¯ is perpendicular to AB¯.

By the theorem, “A radius that is perpendicular to the chord bisects the chord”, we get OQ¯ bisects AB¯.

Consider, ΔOAQ.

Since AB¯ is bisected by OQ¯ and we have AB=6.

AE=3.

In right ΔOAE,

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