   Chapter 6.1, Problem 34E 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

In Exercises 32 to 37, write a paragraph proof.Congruent chords are located at the same distance from the centre of a circle.

To determine

To prove:

The statement “Congruent chords are located at the same distance from the center of a circle”.

Explanation

Given:

The provided statement is “Congruent chords are located at the same distance from the center of a circle”

Theorem used:

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

Hypotenuse leg theorem:

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

Proof:

Consider a circle with center O.

Let AC¯ and DF¯ are the congruent chords. Then AC¯=DF¯

Construct two line segments OB¯ and OE¯ perpendicular to each chord AC¯ and DF¯ respectively as shown below

We know that, the radius that is perpendicular to the chord bisects the chord.

Therefore EF¯=12DF¯ and BC¯=12AC¯.

Since we have AC¯=DF¯, we get EF¯=BC¯

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