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3. Rigid Motions
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Geometry

3.02 Reflections

Lesson
Practice
Lesson

Concept summary

A reflection across the line of reflection is a transformation that produces the mirror image of a geometric figure.

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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.

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.

Line of reflection

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

The main 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)

A figure has reflection symmetry if one half of the figure is the reflection of the other. This is equivalent to there being a line of reflection which maps a figure onto itself. A figure with reflection symmetry is shown below.

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We can see that the triangle's reflection across the line of reflection x=1 is just itself. This means that the triangle has reflection symmetry.

Worked examples

Example 1

For the following graph:

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a

Identify the line of reflection.

Approach

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.

Solution

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.

Approach

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})

Solution

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

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

Reflection

There are multiple ways of representing the same transformation.

Example 2

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

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Solution

The mapping when reflected across the line y=-x is (x,y) \to (-y,-x) Substituting each point into the mapping we get:

  • 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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Example 3

Determine the lines of reflection that map the square in the coordinate plane onto itself.

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Approach

A square has four lines of reflectional symmetry, two connecting the opposite corners and two connecting the midpoints of the opposite sides.

Solution

The lines of symmetry for the square connect the following pairs of points:

  • \left(1,1\right) and \left(3,3\right)
  • \left(1,3\right) and \left(3,1\right)
  • \left(2,1\right) and \left(2,3\right)
  • \left(1,2\right) and \left(3,2\right)

which correspond to the lines

  • y=x
  • y=4-x
  • x=2
  • y=2

Reflection

If we sketch each of these four lines, we can see that the image of the square would be itself if we reflected across any of the lines. This confirms that they are lines of symmetry for the square.

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Outcomes

G.CO.A.1

Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.

G.CO.A.2

Given a rectangle, parallelogram, trapezoid, or regular polygon, determine the transformations that carry the shape onto itself and describe them in terms of the symmetry of the figure.

G.CO.A.3

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

G.CO.A.4

Given a geometric figure, draw the image of the figure after a sequence of one or more rigid motions, by hand and using technology. Identify a sequence of rigid motions that will carry a given figure onto another.

G.MP1

Make sense of problems and persevere in solving them.

G.MP2

Reason abstractly and quantitatively.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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