Chapter 9.1, Problem 49HP

To determine

To find: The equation of the line of reflection.

Answer to Problem 49HP

The equation of the line of reflection is $y=-\frac{5}{3}x+4$ .

Explanation of Solution

Given information:

The expression $A\left(4,3\right)$ is reflected in a line, its image is $A\text{'}\left(-1,0\right)$ .

Calculation:

Calculate the midpoint between $A$ and $A\text{'}$ . This point will be on the line of reflection.

  $\begin{array}{c}\left(\frac{4+(-1)}{2},\frac{3+0}{2}\right)\\ =\left(\frac{3}{2},\frac{3}{2}\right)\end{array}$

Calculate the slope between $A$ and $A\text{'}$ . The slope of the line of the reflection is the opposite reciprocal.

Slope $AA\text{'}=\frac{0-3}{-1-4}=\frac{3}{5}$

Slope of reflection line $-\frac{5}{3}$

Use point slope form of all equation to write the equation of the line of reflection. $y-y_1=m\left(x-x_1\right)$

  $\begin{array}{c}y-\frac{3}{2}=-\frac{5}{3}\left(x-\frac{3}{2}\right)\\ y-\frac{3}{2}=-\frac{5}{3}x+\frac{5}{2}\\ y=-\frac{5}{3}x+4\end{array}$

Therefore, the equation of the line of reflection is $y=-\frac{5}{3}x+4$ .