Exercise 13 Page 558

To reflect a polygon across a line, we will follow three steps.

1. Graph the preimage and identify the line of reflection on the coordinate plane.
2. Locate the image of each vertex so that the reflection line is the perpendicular bisector of the segment whose endpoints are the vertex and its image.
3. Connect the new vertices to draw the image of the given figure.

Let's do it!

Step $1$

We will begin by plotting the given vertices and graphing the figure. Since the given reflection is $R_{x\text{-axis}},$ the line of reflection is the $x\text{-}$axis. Note that the equation of this line is $y=0.$

Step $2$

Now, we will locate the image of each vertex so that the $x\text{-}$axis is the perpendicular bisector of the segment whose endpoints are the vertex and its image. The distance between a preimage vertex and the $x\text{-}$axis must be the same as the distance between its corresponding image vertex and the $x\text{-}$axis. This is shown for vertex $A$ below.

If you need further explanation on how to draw a line that passes through a given point and is perpendicular to a given line, please refer to this explanation.

Step $3$

Finally, we can connect the new vertices to draw the reflection image of the given figure.