Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10: 9781285195698
ISBN 13: 978-1-28519-569-8

Chapter 2 - Section 2.2 - Indirect Proof - Exercises - Page 81: 31


$M$ 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 $\overline{AB}$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.