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 - Review Exercises - Page 495: 33

Answer

If the diagonals of a given rectangle are perpendicular, the shape is a square.

Work Step by Step

We name the vertices: $A:(0,0) \\ B: (0,b) \\ C: (a,b) \\ D: (a,0)$ We see if a is equal to b. Since the lines are perpendicular, it follows that the product of their slopes is $-1$: $ m_1= \frac{b-0}{a-0} = \frac{b}{a}$ $ m_2= \frac{b-0}{0-a} = \frac{b}{-a}$ Thus: $\frac{b}{-a} \times \frac{b}{a} = -1$ This is only true when a and b are equal, so we see that the shape is a square.
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