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 10 - Review Exercises - Page 495: 32


When the midpoints of the sides of a parallelogram are connected, the shape formed is a parallelogram.

Work Step by Step

We draw the base of the parallelogram on the x-axis, with a vertex at (0,0). We define the midpoints as follows: $ B: (b,c) \\ C: (a+2b,2c) \\ D: (2a+b,c) \\E: (a,0)$ We use the equation for slope: $m = \frac{y_2-y_1}{x_2-x_1}$ We find: $m_1 = \frac{2c-c}{2b+a-b}=\frac{c}{b+a}$ $m_2 = \frac{2c-c}{a+2b-2a-b}=\frac{b-a}{c}$ $m_3 = \frac{c-0}{2a+b-a}=\frac{c}{b+a}$ $m_4 =\frac{c}{b-a}$ We see that the slopes of opposite sides are equal, making opposite sides parallel and making the shape a parallelogram.
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