   Chapter 6.1, Problem 18E 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

A B ¯ is the common chord of ⊙ O and ⊙ Q . If A B = 12 and each circle has a radius of length 10 , how long is O Q ¯ ? Exercises 18, 19

To determine

To calculate:

The length of OQ¯.

Explanation

Given:

AB¯ is the common chord of O and Q. If AB=12 and each circle has a radius of length 10.

Theorem used:

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

2. An angle inscribed in a semicircle is a right angle.

Calculation:

In the circle O, OA¯ and OB¯ are radii and in circle Q, QA¯ and QB¯ are radii. By joining these radii we get a rhombus with sides of length 10.

Thus AQBO is a rhombus with diagonals OQ¯ and AB¯.

We know that the diagonals of a rhombus 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=12.

AE=6

Now consider two triangles ΔOAE and ΔQAE.

Here, OA¯QA¯, EA¯EA¯ and AEOAEQ

ΔOAEΔQAE

Since, the two triangles are congruent, we get OE¯QE¯

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