Chapter 4.3, Problem 3CYU

To determine

To find: Whether the paths AC and BD of the garden are perpendicular or not. Explain.

Answer to Problem 3CYU

The paths AC and BD of the garden are perpendicular to each other.

Explanation of Solution

Given information: A garden is in the shape of a quadrilateral with given vertices, $A\left(-2,1\right),B\left(3,-3\right),C\left(5,7\right)\text{ and }D\left(-3,4\right)$ respectively. And two paths along AC and BD cut across the garden at some point.

Calculation: The paths AC and BD of the garden will be perpendicular if slope $\left(m_{AC}\right)$ of the path AC and slope $\left(m_{BD}\right)$ of the path BD of the garden are negative reciprocal to each other. Therefore, evaluating slope $\left(m_{AC}\right)$ of the path AC and slope $\left(m_{BD}\right)$ of the path BD of the garden respectively.

  $\begin{array}{c}{m}_{AC}=\frac{7-1}{5+2}\\ =\frac{6}{7}\mathrm{......}\left(1\right)\\ m_{BD}=\frac{4+3}{-3-3}\\ =-\frac{7}{6}\mathrm{......}\left(2\right)\end{array}$

From (1) and (2) are negative reciprocal ton each other. Therefore, the paths AC and BD of the garden are perpendicular to each other.