Chapter 4.3, Problem 3CYU

To determine

To verify the perpendicularity of given paths.

Answer to Problem 3CYU

The paths are perpendicular.

Explanation of Solution

Given:

A garden is in the shape of a quadrilateral with vertices

  $A\left(-2,1\right),B\left(3,-3\right),C\left(5,7\right)\&D\left(-3,4\right)$ . Two paths represented by $\overline{AC}$ and $\overline{BD}$ cut across

the garden.

Concept used:

Two lines are perpendicular if and only if the product of their slopes is $-1$ .

Slope of a line joining the points $\left(x_1,y_1\right)\&\left(x_2,y_2\right)$ is $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

Let m1 and m2 be the slopes of the paths $\overline{AC}$ and $\overline{BD}$ respectively.

Now slope of the line joining the points $A\left(-2,1\right),C\left(5,7\right)$ is,

  $m_1=\frac{7-1}{5-\left(-2\right)}=\frac{6}{7}$

Similarly the slope of the line joining the points $B\left(3,-3\right),D\left(-3,4\right)$ is,

  $m_2=\frac{4-\left(-3\right)}{-3-3}=-\frac{7}{6}$

Now,

  $m_1\cdot m_2=\frac{6}{7}\cdot \left(-\frac{7}{6}\right)=-1$

Therefore, the paths are perpendicular.