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


When the midpoints of a parallelogram are connected, they form a rectangle.

Work Step by Step

We name the sides of the rhombus: $(0,0) ; (2b,2c); (2a+2b, 2c); (2a,0)$ Thus, the midpoints are: $(b,c); (a+2b, 2c); (2a+b,c); (a,0)$ Plugging these into the distance formula, we find: $s_1 = \sqrt{c^2 +(b-a)^2} $ $s_2 = \sqrt{c^2 +(-b-a)^2} $ $s_3 = \sqrt{c^2 +(-b-a)^2} $ $s_4 = \sqrt{c^2 +(b-a)^2} $ Since opposite sides are congruent and the angles are 90 degrees, it follows that the shape is a rectangle.
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