Topics
6. 2D Geometry
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United States of AmericaVirginia
Grade 8

6.07 Translations and reflections in the coordinate plane

Lesson
Practice

Combinations of translations and reflections

Recall, a translation is a transformation where an image is formed by moving every point on the preimage the same distance in the same direction.

A reflection is a transformation where an image is formed by 'flipping' the preimage over a line called the line of reflection. Each point on the image is the same distance from the line of reflection as the corresponding point in the preimage.

We've practiced translations and reflections separately, but these transformations can also be combined. By combining translations and reflections, we can achieve a wide variety of movements and create interesting patterns or solve practical problems.

Exploration

Use the graph of polygon ABCDE to explore combining translations and reflections.

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Apply each of the following pairs of transformations and identify the coordinates of the image:

  • Reflect polygon ABCDE over the x-axis. Then, translate the polygon up 3 units.
  • Translate polygon ABCDE up 3 units. Then, reflect the polygon over the x-axis.

1. Did changing the order of the transformations affect the image?

  • Reflect polygon ABCDE over the y-axis. Then, translate the polygon up 6 units.
  • Translate polygon ABCDE up 6 units. Then, reflect the polygon over the y-axis.

2. Did changing the order of the transformations affect the image?

  • Choose your own translation and reflection and test your hypothesis.

Translating a shape first and then reflecting it across the x- or y-axis may not give the same result as reflecting it first and then translating it.

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The graph translates \triangle ABC to the left 4 units and down 2 units to create \triangle A'B'C'. Then, \triangle A'B'C' is reflected over the x-axis to create \triangle A''B''C''.

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The order of those same two transformations are reversed. \triangle ABC is reflected over the x-axis and then translated left 4 units and down 2 units.

The final image \triangle{A''B''C''} is in a different location than before. Changing the order of transformations, changed the image.

Let's try a new translation and a reflection on \triangle ABC.

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First, let's translate the triangle up 2 units. Then, reflect the triangle over the y-axis.

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First, reflect the triangle over the y-axis. Then, translate the triangle up 2 units.

The final image \triangle{A''B''C''} is in the same location as before. Changing the order of transformations did not change the image.

Examples

Example 1

Consider \triangle ABC and \triangle A''B''C''.

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Describe the combination of transformations to get from the preimage to the image.

Worked Solution
Create a strategy

We will use some combination of reflections and translations to align the two shapes exactly. There are multiple correct solutions.

Apply the idea

Find some combination of transformations to align the two shapes.

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Reflect \triangle ABC across the x-axis, to get\triangle A'B'C'.

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Then translate \triangle A'B'C', 10 units to the left and 8 units down, to get \triangle A''B''C''.

Reflect and check

Find a new combination of transformations to align the two shapes.

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Translate \triangle ABC, 10 units to the left and 8 units up to get\triangle A'B'C'.

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Then reflect \triangle A'B'C'over the x-axis to get \triangle A''B''C''.

Example 2

Use \triangle RST to answer the following questions.

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a

Translate \triangle RST down 3 units and left 2 units. Then, reflect the triangle over the y-axis. Graph the image on the same coordinate plane as the preimage of \triangle RST.

Worked Solution
Create a strategy

A translation shifts the every point on the triangle the same distance in the same direction. Then, a reflection over the y-axis flips the triangle onto the other side of the y-axis.

Apply the idea

First, perform the translation on the graph down 3 units and left 2 units to create \triangle R'S'T'.

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Then, reflect \triangle R'S'T' over the y-axis.

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Reflect and check

We can check the ordered pairs algebraically.

The first transformation translated \triangle RST down 3 units and left 2 units. We can subtract 3 from each y-coordinate and subtract 2 from each 2 from each x-coordinate to find the ordered pairs for \triangle R'S'T'.

R(2,\,3) \to R'(2-2,\,3-3) \to R'(0,\,0)

S(5,\,6) \to S'(5-2,\,6-3) \to S'(3,\,3)

T(1,\,8) \to T'(1-2,\,8-3) \to T'(-1,\,5)

The second transformation reflected \triangle R'S'T' over the y-axis. This reflection does not change any of the y-coordinates, but all of the x-coordinates will change signs to form the ordered pairs for \triangle R''S''T''.

R'(0,\,0) \to R''(0,\,0)

S'(3,\,3) \to S''(-3,\,3)

T'(-1,\,5) \to T''(1,\,5)

b

If the order of those transformations are reversed, would it result in the same image?

Worked Solution
Create a strategy

Transform \triangle RST in reverse order to see if the same image is created.

Apply the idea

First we will reflect \triangle RST over the y-axis to create \triangle R'S'T'.

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Then, we translate \triangle R'S'T' down 3 units and left 2 units to create \triangle R''S''T''.

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When the order of the transformations is reversed, it results in a different image.

Example 3

A local community is planning to revamp an old park by introducing a creative playground shaped as a polygon. The current design proposal includes a quadrilateral shaped play area with coordinates at P(2,\,3),\,Q(5,\,5),\,R(5,\,1), and S(3,\,1) on a grid system that represents a scaled down version of the park. To integrate this new structure seamlessly with the park's layout, two specific transformations are required:

  1. Mirror the proposed playground layout across the y-axis to better align with other park facilities.

  2. The mirrored layout should be moved 3 units downward and 4 units to the right to fit precisely in the designated area for playgrounds.

Identify the final coordinates of the vertices after applying these transformations.

Worked Solution
Create a strategy

To reflect a point across the y-axis, we change the sign of the x-coordinate while keeping the y-coordinate the same.

To translate a point, add the translation values to the coordinates.

Apply the idea

Reflecting each vertex:

P(2,\,3) \longrightarrow P'(-2,\,3)

Q(5,\,3) \longrightarrow Q'(-5,\,3)

R(5,\,1) \longrightarrow R'(-5,\,1)

S(3,\,1) \longrightarrow S'(-3,\,1)

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After reflecting across the y-axis, the vertices are:P'(-2,3),\,Q'(-5,3),\,R'(-5,1),\,S'(-3,1)

Translating each mirrored vertex:

P'(-2,\,3) \longrightarrow P''(2,\,0)

Q'(-5,\,3) \longrightarrow Q''(-1,\,0)

R'(-5,\,1) \longrightarrow R''(-1,\,-2)

S'(-3,\,1) \longrightarrow S''(1,\,-2)

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After translating, the final coordinates of the vertices are:P''(2,\,0),\,Q''(-1,\,0),\,R''(-1,\,-2),\,S''(1,\,-2)

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Idea summary

When we translate and then reflect a shape over the x- or y-axis, it may not give the same result as when we reflect and then translate.

If we translate, and then reflect:

  1. Translation: Move the shape to a new location.
  2. Reflection: Flip the moved shape over the x- or y-axis.
  3. Result: The shape's final position depends on where it was moved before being flipped.

If we reflect, and then translate:

  1. Reflection: Flip the shape first over the x- or y-axis.
  2. Translation: Move the flipped shape to a new location.
  3. Result: The shape's final position depends on where it was flipped before being moved.

Outcomes

8.MG.3

The student will apply translations and reflections to polygons in the coordinate plane.

8.MG.3c

Given a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been translated and reflected over the x- or y-axis or reflected over the x- or y-axis and then translated.

8.MG.3d

Sketch the image of a polygon that has been translated vertically, horizontally, or a combination of both.

8.MG.3e

Sketch the image of a polygon that has been reflected over the x- or y-axis.

8.MG.3f

Sketch the image of a polygon that has been translated and reflected over the x- or y-axis, or reflected over the x- or y-axis and then translated.

8.MG.3g

Identify and describe transformations in context (e.g., tiling, fabric, wallpaper designs, art).

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