Chapter 13.3, Problem 18PSC

To determine

To find: The reflection of the point over the given graph.

Answer to Problem 18PSC

The reflection of the point $\left(-6,5\right)$ is $\left(-2,-3\right)$ .

Explanation of Solution

Given:

The given equation of graph is:

  $2y-x=6\left(1\right)$ .

The given point of which reflected need to find is $\left(-6,5\right)$ .

Calculation:

The equation $\left(1\right)$ can be written as:

  $\begin{array}{l}y=\frac{1}{2}x+3\\ \text{slope}=\frac{1}{2}\end{array}$

The line joining the point and its reflection is perpendicular to the given line, so it has a slope is $-2$ .

The equation of line joining the point and its reflection is:

  $\begin{array}{l}y=-2x+b\\ 5=-2\left(-6\right)+b\because \text{line passes through}\left(-6,5\right)\\ b=-7\\ y=-2x-7\left(2\right)\end{array}$

Further simplified as:

  $\begin{array}{c}\frac{1}{2}x+3=-2x-7\\ x=-4\\ y=2\left(-4\right)-7\left[\because \text{substitute }x=-4\text{in equation}\left(2\right)\right]\\ y=1\end{array}$

  $\left(-4,1\right)$ is the mid-point of $\left(-6,5\right)$ and $\left(x,y\right)$ .

Here $\left(x,y\right)$ is the reflection point of reflection of the point.

The mid-point is calculated as:

  $\begin{array}{c}\left(\frac{-6+x}{2},\frac{5+y}{2}\right)=\left(-4,1\right)\\ \frac{-6+x}{2}=-4\\ \frac{5+y}{2}=1\\ -6+x=-8\end{array}$

Further simplified as:

  $\begin{array}{c}x=-2\\ 5+y=2\\ y=-3\end{array}$

Therefore the reflection of the point $\left(-6,5\right)$ is $\left(-2,-3\right)$ .