Question

each figure in the diagram at the right is a reflection of another figure across one of the reflection lines. a. write a reflection rule to describe figure 1. b. write a reflection rule to describe figure 2. c. write a reflection rule to describe figure 4. d. justify your answers for parts (a)-(c). figure 1 = r (figure )

Explanation:

Step1: Recall reflection rule concept

In a coordinate - plane, a reflection over the x - axis has the rule $(x,y)\to(x, - y)$ and over the y - axis has the rule $(x,y)\to(-x,y)$. Without seeing the actual figure and its position relative to the reflection line, assume a general case. If we consider a reflection over the y - axis for Figure 1.
Let a point on Figure 1 be $(x,y)$. After reflection over the y - axis, the new point is $(-x,y)$. So the reflection rule for Figure 1 (assuming reflection over y - axis) is $(x,y)\to(-x,y)$.

Step2: For Figure 2

If we assume it is reflected over the x - axis. Let a point on Figure 2 be $(x,y)$. After reflection over the x - axis, the new point is $(x,-y)$. So the reflection rule for Figure 2 is $(x,y)\to(x, - y)$.

Step3: For Figure 4

Suppose it is reflected over a vertical line $x = a$. Let a point on Figure 4 be $(x,y)$. The distance between the point $(x,y)$ and the line $x=a$ is $|x - a|$. The new x - coordinate after reflection over the line $x = a$ is $2a - x$ and the y - coordinate remains the same. So the rule is $(x,y)\to(2a - x,y)$.

Step4: Justification

The reflection rules are based on the geometric property that the reflection line is the perpendicular bisector of the line segment joining a point and its image. For a reflection over the x - axis, the x - coordinate stays the same and the y - coordinate changes sign because the distance from the x - axis is measured in the opposite direction. For a reflection over the y - axis, the y - coordinate stays the same and the x - coordinate changes sign. For a reflection over a vertical line $x = a$, we use the concept of equal distances from the line of reflection.

Answer:

a. $(x,y)\to(-x,y)$ (assuming reflection over y - axis)
b. $(x,y)\to(x, - y)$ (assuming reflection over x - axis)
c. $(x,y)\to(2a - x,y)$ (assuming reflection over line $x = a$)
d. Justified as above.