Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Appendix B - Coordinate Geometry and Lines - B Exercises: 13

Answer

The points $(-2,9)$, $(4,6)$ and $(1,0)$, $(-5,3)$ are vertices of a square.

Work Step by Step

We are given the square with vertices $A(-2,9)$, $B(4,6)$ and $C(1,0)$, $D(-5,3)$. Consider AB, BC , CD and AD are four sides of a square. Use distance formula $d=\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$ and slope formula $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ For side AB: $AB=\sqrt {(4-(-2))^{2}+(6-9)^{2}}=\sqrt {36+9}=\sqrt {45}$ $m=\frac{6-9}{4-(-2)}=-\frac{1}{2}$ For side BC: $BC=\sqrt {(1-4)^{2}+(0-6)^{2}}=\sqrt {9+36}=\sqrt {45}$ $m=\frac{0-6}{1-4}=2$ For side CD: $CD=\sqrt {(-5-1)^{2}+(3-0)^{2}}=\sqrt {36+9}=\sqrt {45}$ $m=\frac{3-0}{-5-1}=-\frac{1}{2}$ For side AD: $AD=\sqrt {(-5-(-2))^{2}+(3-9)^{2}}=\sqrt {9+36}=\sqrt {45}$ $m=\frac{3-9}{-5-(-2)}=2$ All sides are equal and their consecutive sides are perpendicular having negative reciprocal of their slope, therefore, the given points are vertices of a square. Hence, the points $(-2,9)$, $(4,6)$ and $(1,0)$, $(-5,3)$ are vertices of a square.
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