   Chapter 6.CR, Problem 13CR 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

Two circles are concentric. A chord of the larger circle is also tangent to the smaller circle. The lengths of the radii are 20 and 16, respectively. Find the length of the chord.

To determine

To find: The length of the chord of the larger circle.

Explanation

Theorem:

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

2) If a line through the centre of a circle bisects a chord other than a diameter, then it is perpendicular to the chord.

Calculation:

Given that the length of the radius of a larger circle is 20.

The length of the radius of a smaller circle is 16.

Let O be the center of the two circles and AB be a chord.

Also, AB be a tangent to the smaller circle at C

So, AB = 24 mm and OB = radius = 15 mm

The distance from the center of the larger circle to the chord = OC.

By theorem, OCAB.

So, OCB is a right triangle.

By theorem, AC=CB or AB = 2AC = 2CB.

So, AC=BC=12AB.

So, OC=16 and OB=20

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