Elementary Geometry for College Students (7th Edition)

Published by Cengage
ISBN 10: 978-1-337-61408-5
ISBN 13: 978-1-33761-408-5

Chapter 2 - Section 2.2 - Indirect Proof - Exercises - Page 94: 33


$M$ is the only midpoint for $\overline{AB}$.

Work Step by Step

If $M$ is a midpoint of $\overline{AB}$, then $AM$ = $\frac{1}{2}AB$. Assume that $N$ is also a midpoint of $AB$ so that $AN$ = $\frac{1}{2}AB$. By that substitution, $AM$ = $AN$. By the segment-Addition Postulate, $AM$ = $AN$ =$NM$. Using substitution again, $AN$ + $NM$ = $AN$. Subtracting Gives $NM$ = $0$. But this contradicts the Ruler Postulate, which states tha the measure of a line segment is a positive number. Therefore, our assumption is wrong and $M$ is the only midpoint for $\overline{AB}$.
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