Elementary Geometry for College Students (7th Edition) Clone

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

Chapter 10 - Section 10.4 - Analytic Proofs - Exercises - Page 472: 7

Answer

The midpoints of quadrilaterals form a parallelogram.

Work Step by Step

We assign coordinates for each point on quadrilateral ABCD: $A: 0,0 \\ B: 2a, 2b \\ C: 2c, 2d \\ D: 2e, 0$ This means that the midpoints are: $(a,b); (a+c, b+d); (e,0); (e+c, d)$ Thus, the slopes are: $m_1 = \frac{b+d-b}{a+c-c} =\frac{d}{c}$ $m_2 = \frac{b+d-d}{a+c-e-c} =\frac{b}{a-e}$ $m_3 = \frac{d-0}{e+c-e} =\frac{d}{c}$ $m_4 = \frac{b-0}{a-e} =\frac{b}{a-e}$ Since corresponding slopes are equal, the figure inside of the quadrilateral is a parallelogram.
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