Topics
4. Rigid Motions
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United States of America
Grades 9-12

4.02 Reflections

Lesson
Practice

Introduction

Reflections are a part of the rigid transformations studied in 8th grade. We will use slopes and distances to verify reflections in this lesson and solidify our understanding of reflected figures in the coordinate plane.

Reflections

Exploration

Drag the points to create a triangle. Check the boxes to show the line of reflection, image, and movement of points. Drag the sliders to change the line of reflection.

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  1. What happens to the image as you move the line of reflection?
  2. What do you notice about the angle formed by the segments from the pre-image to the image and the line of reflection?
  3. What can we say about the distance of each figure to the line of reflection?

We can think of a reflection as a function which sends the input point to an output point such that the line of reflection is the perpendicular bisector of the two points.

Reflection

A transformation that produces the mirror image of a figure across a line

Line of reflection

A line that a figure is flipped over to create a mirror image

In other words, the line of reflection is always the perpendicular bisector of the line segment joining corresponding points in the pre-image and image. Because of this, the line of reflection will always be equidistant from the two corresponding points in the pre-image and image, so we get a mirror image over the line of reflection.

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The function notation for a reflection of a shape: R_{\text{line of reflection}}(\text{shape}). For example, if we wanted to reflect polygon ABCD over the line y=2x, we would write R_{y=2x}\left(ABCD\right) = A'B'C'D'.

The most common lines of reflection have the following impact on a point:

  • Line of reflection: x-axis \qquad Transformation mapping: \left(x, y\right) \to \left(x, -y \right)

  • Line of reflection: y-axis \qquad Transformation mapping: \left(x, y\right) \to \left(-x, y \right)

  • Line of reflection: y=x \, \, \qquad Transformation mapping: \left(x, y\right) \to \left(y, x \right)

  • Line of reflection: y=-x \, \, \, \quad Transformation mapping: \left(x, y\right) \to \left(-y, -x \right)

Examples

Example 1

For the following graph:

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a

Identify the line of reflection.

Worked Solution
Create a strategy

The line must be equal distance between the pairs A and A', B and B', and C and C'. So to find the line of reflection, we find the average between each pair and then find the line that goes through all the averages.

Apply the idea

The average of A and A': \left(7,0\right)

The average of B and B': \left(-8,0\right)

The average of C and C': \left(8,0\right)

The line of reflection goes through all of the above points. Therefore the line of reflection is the x-axis, y=0.

b

Write the transformation mapping in both coordinate and function notation.

Worked Solution
Create a strategy

The reflection mapping is given either by the change in arbitrary point \left(x,y\right), or by using function notation R_{\text{line of reflection}}(\text{shape})

Apply the idea

Coordinate notation: \left(x,y\right) \to \left(-x,y\right)

Function notation: R_{y=0}\left(\triangle ABC\right) = \triangle A'B'C'

Example 2

Determine the image of the quadrilateral PQRS when reflected across the line y=-x.

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Worked Solution
Create a strategy

Recall the rule for reflecting across the line y=-x is \left(x,y\right) \to \left(-y,-x\right)

Apply the idea

Reflecting each point, we get the following mapping:

  • The point P\left(1,1\right) is reflected to P'\left(-1,-1\right)
  • The point Q\left(3,3\right) is reflected to Q'\left(-3,-3\right)
  • The point R\left(3,2\right) is reflected to R'\left(-2,-3\right)
  • The point S\left(2,0\right) is reflected to S'\left(0,-2\right)

Plotting these points and connecting them gives us the image:

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Reflect and check

To visualize the reflection, we could use a mirror or tracing paper to map each vertex of the pre-image to its image across the line. These tools are also useful if we forget the mapping rule.

Idea summary

The most common lines of reflections have the following impact on a point:

  • Line of reflection: x-axis \qquad Transformation mapping: \left(x, y\right) \to \left(x, -y \right)

  • Line of reflection: y-axis \qquad Transformation mapping: \left(x, y\right) \to \left(-x, y \right)

  • Line of reflection: y=x \, \, \qquad Transformation mapping: \left(x, y\right) \to \left(y, x \right)

  • Line of reflection: y=-x \, \, \, \quad Transformation mapping: \left(x, y\right) \to \left(-y, -x \right)

Outcomes

G.CO.A.2

Represent transformations in the plane using, e.g. Transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. Translation versus horizontal stretch).

G.CO.A.4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G.CO.A.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. Graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

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