Using the GeoGebra applet, observe polygon ABCD and its reflected image A′B′C′D′across the line . Use the slider to change the slope mm of the line. Your goal is to determine how the reflected polygon changes and to derive the general rule for reflecting any point over the line
A. Observation & Exploration
Set the slope to m = 1 (line . Observe the vertices of ABCD and A′B′C′D′. What is the relationship between the coordinates of a vertex and its reflected image?
Move the slider so the line becomes y = −x. Now how are the coordinates of each vertex related to its reflected image?
As you slowly change m with the slider, describe how the reflected polygon moves. What stays the same about the original polygon and its reflection, regardless of the slope?
Synthesis & Explanation
As you vary the slope m , describe how the orientation of the reflected polygon changes relative to the original. What happens when m = 0 or when the line is very steep?
Synthesis & Explanation
Reflection is a "rigid motion." What properties of the original polygon ABCDABCD (side lengths, angles, area) are preserved in A′B′C′D′A′B′C′D′? How does the applet demonstrate this?
Generalization
In your own words, describe the two main geometric rules that define a reflection over any line