Questions: What is the reflection rule that maps the triangle to its image? P(8,4), Q(-6,5), R(4,5) and P'(8,-12), Q'(-6,-13), R'(4,-13) Ry--x Ry-A Rx-4 Ry-x

Transcript text: What is the reflection rule that maps the triangle to its image? \[ P(8,4), Q(-6,5), R(4,5) \text { and } P^{\prime}(8,-12), Q^{\prime}(-6,-13), R^{\prime}(4,-13) \] $R_{y--x}$ $R_{y-A}$ $R_{\mathrm{x}-4}$ $R_{y-x}$

Solution

Solution Steps

To determine the reflection rule that maps the triangle to its image, we need to analyze the coordinates of the original points and their corresponding reflected points. By comparing the coordinates, we can identify the axis or line of reflection.

Solution Approach

Compare the y-coordinates of the original points and their images to see if they are negated or shifted.
Compare the x-coordinates of the original points and their images to see if they are negated or shifted.
Determine the line of reflection based on the changes observed in the coordinates.

Step 1: Identify Original and Reflected Points

The original points of the triangle are given as:

$ P(8, 4) $
$ Q(-6, 5) $
$ R(4, 5) $

The reflected points are:

$ P'(8, -12) $
$ Q'(-6, -13) $
$ R'(4, -13) $

Step 2: Analyze the Reflection

To determine the reflection rule, we compare the coordinates of the original points with their corresponding reflected points.

The x-coordinates of points $ P $, $ Q $, and $ R $ remain unchanged in their reflected counterparts $ P' $, $ Q' $, and $ R' $.
The y-coordinates change as follows:
- For $ P $: $ 4 $ to $ -12 $ (a change of $ -16 $)
- For $ Q $: $ 5 $ to $ -13 $ (a change of $ -18 $)
- For $ R $: $ 5 $ to $ -13 $ (a change of $ -18 $)

This indicates that the reflection occurs over a vertical line, as the x-coordinates are constant while the y-coordinates are transformed.

Step 3: Determine the Line of Reflection

The consistent x-coordinate suggests that the reflection is over a vertical line. The y-coordinates indicate a downward shift, which can be expressed as a reflection over the line $ x = a $.