Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 6 - Section 6.1 - Circles and Related Segments and Angles - Exercises - Page 277: 34

Answer

Let us consider two chords that are congruent. By definition, when we draw the radius through each chord at a 90 degree angle, each chord will be bisected. Now, we draw the radius from the center of the circle to one end of each chord. This will form a right triangle, with half of each chord being a leg of a triangle, the radius bisecting each chord being the other leg of the triangle, and the radius intersecting the end of the chord being the hypotenuse. Since each segment is congruent and is bisected, each of the legs created by the bisector are congruent. In addition, since radius for a circle does not change, the hypotenuse of each triangle is congruent. Thus, by the Pythagorean theorem, the third side of these triangles are congruent, meaning that they are equal distances from the center of the circle.
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