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

6.05 Translations in the coordinate plane

Lesson
Practice

Translations in the coordinate plane

A transformation of a figure is a mapping that changes the size, shape, and/or position of the figure. The original figure is called the preimage and the new figure created after applying the transformation(s) is called the image.

Exploration

This applet shows the translation of an object to its image. You can use the sliders to change the horizontal and vertical amounts.

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  1. How did the object change as you changed the horizontal slider on the applet?

  2. How did the object change as you changed the vertical slider on the applet?

  3. What remained the same between the object and image as you used either or both of the horizontal and vertical sliders?

A transformation of preimage point A creates an image that can be named using the notation A' (read as "A prime").

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A translation is what occurs when we move an object or shape from one place to another without changing its size, shape or orientation.

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To identify and describe a translation

  1. Look for corresponding points on the object

  2. Identify if the object has moved horizontally, vertically, or both and then describe that movement as left/right or up/down

  3. Count the number of places (or units) that the object has moved

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When applied to a figure, a translation moves every point on an object or shape the same distance in the same direction.

Translations maintain congruence between the preimage and image but change the location on the coordinate plane.

This figure has been translated horizontally, 4 units to the left.

Examples

Example 1

What is the translation from triangle ABC to triangle A'B'C'?

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Worked Solution
Create a strategy

Consider just one vertex from triangle ABC and the corresponding vertex from triangle A'B'C'. Count the units between each vertex and consider the direction of movement.

Apply the idea

Consider vertices C and C'. We count 7 units up from C to C'.

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Triangle ABC is translated 7 units up to form triangle A'B'C'.

Reflect and check

We count 7 units up from A to A'.

We count 7 units up from B to B'.

We can count 7 units up from any point on triangle ABC to the triangle A'B'C'. For example, the point \left(7,-5\right) on triangle ABC corresponds to point \left(7,2\right) on triangle A'B'C'.

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Example 2

What is the translation from quadrilateral A to quadrilateral B?

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Worked Solution
Create a strategy

Identify the directions of the movements of the corresponding vertices and count the number of steps.

Apply the idea

We can see that the vertices moved 4 units right and 2 units up.

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So the translation is 4 units right and 2 units up.

Reflect and check

We count 4 units right and 2 units up from any point on A to any point on B.

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Example 3

Quadrilateral PQRS is to be translated 3 units to the left and 2 units down.

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a

Determine the coordinates of vertex P' of the translated shape.

Worked Solution
Create a strategy

Vertex P needs to be translated 3 units to the left and 2 units down.

Apply the idea

Count 3 units to the left and 2 units down from P to form P'.

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Vertex P' is at \left( 1 , 0\right).

Reflect and check

We can check the ordered pair by subtracting 3 units from the x-coordinate and subtracting 2 units from the y-coordinate.

Vertex P is at \left(4, 2\right), so vertex P' is at \left(4-3, 2-2\right)=\left(1, 0\right)

b

Sketch the original and translated quadrilateral PQRS and P'Q'R'S' on the same coordinate plane.

Worked Solution
Create a strategy

Every point on quadrilateral PQRS is translated 3 units to the left and 2 units down.

Apply the idea

Count 3 units to the left and 2 units down for each vertex on PQRS to form P'Q'R'S'.

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Reflect and check

We can check each ordered pairs by subtracting 3 units from each x-coordinate and subtracting 2 units from each y-coordinate.

Vertex Q is at \left(4, 6\right), so vertex Q' is at \left(4-3, 6-2\right)=\left(1, 4\right)

Vertex R is at \left(7, 6\right), so vertex R' is at \left(7-3, 6-2\right)=\left(4, 4\right)

Vertex S is at \left(7, 2\right), so vertex S' is at \left(7-3, 2-2\right)=\left(4, 0\right)

Example 4

You're helping a friend navigate a theme park using a map on a coordinate plane. The starting point is the fountain, located at A\left(1,2\right). The goal is to reach three attractions in a specific order: the roller coaster, the Ferris wheel, and the food court.

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From the fountain, walk 5 units to the right and 3 units down to reach the roller coaster. Identify the coordinates of the roller coaster.

Worked Solution
Create a strategy

First plot the fountain at \left(1,2\right) on the coordinate plane. Then translate it right 5 units and down 3 units.

Apply the idea

Plot the point of the fountain on the graph.

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Then, translate point A to the right 5 units and down 3 units.

Name the point of the roller coaster point B.

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Point B, the location of the roller coaster, is at \left(6, -1\right)

Reflect and check

We can check the ordered pairs by adding 5 units to the x-coordinate and subtracting 3 units from the y-coordinate.

The fountain is at \left(1, 2\right), so the roller coaster is at \left(1+5, 2-3\right)=\left(6, -1\right).

b

From the roller coaster, walk 4 units to the left and 6 units up to reach the Ferris wheel. Identify the coordinates of the Ferris wheel.

Worked Solution
Create a strategy

Translate the ordered pair of the roller coaster left 4 units and up 6 units.

Apply the idea

Translate point B, the roller coaster, to the left 4 units and up 6 units.

Name the point of the Ferris wheel point C.

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Point C, the location of the Ferris wheel, is at \left(2, 5\right)

Reflect and check

We can check the ordered pairs by subtracting 4 units from the x-coordinate and adding 6 units to the y-coordinate.

The roller coaster is at \left(6, -1\right), so the roller coaster is at \left(6-4, -1+6\right)=\left(2, 5\right).

c

From the Ferris wheel, walk 3 units to the right and 2 units down to reach the food court. Identify the coordinates of the food court.

Worked Solution
Create a strategy

Translate the ordered pair of the Ferris wheel right 3 units and down 2 units.

Apply the idea

Translate point C, the Ferris wheel, to the right 3 units and down 2 units.

Name the point of the food court point D.

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Point D, the location of the food court, is at \left(5, 3\right)

Reflect and check

We can check the ordered pairs by adding 3 units to the x-coordinate and subtracting 2 units from the y-coordinate.

The Ferris wheel is at \left(2, 5\right), so the roller coaster is at \left(2+3, 5-2\right)=\left(5, 3\right).

d

Describe the path taken from the fountain to the food court if the friend decides to avoid the roller coaster and the Ferris wheel.

Worked Solution
Create a strategy

Plot the points of the fountain and the food court on a graph. Calculate the distance needed to travel to get from the fountain to the food court on the graph.

Apply the idea

Plot the points of the fountain, point A, and the food court, point D.

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Count the horizontal and vertical distance from point A, the fountain, to point D, the food court.

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To get from the fountain at A\left(1, 2\right) to the food court at D\left(5, 3\right) the friend needs to travel right 4 units and up 1 unit.

Reflect and check

Since we already knew the ordered pairs of the fountain and the food court, we could have found the difference between the points to calculate the translation.

The x-coordinate of the food court is 5 and the x-coordinate of the fountain is 1.

The difference between 5 and 1 is a positive 4. So the friend needs to travel 4 units to the right.

The y-coordinate of the food court is 3 and the y-coordinate of the fountain is 2.

The difference between 3 and 2 is a positive 1. So the friend needs to travel 1 unit up.

Therefore, the complete translation is 4 units to the right and 1 unit up.

Idea summary

To identify and describe a translation

  1. Look for corresponding points on the object

  2. Identify if the object has moved horizontally or vertically, and then describe that movement as left/right or up/down

  3. Count the number of places (or units) that the object has moved

Outcomes

8.MG.3

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

8.MG.3a

Given a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been translated vertically, horizontally, or a combination of both.

8.MG.3d

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

8.MG.3g

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

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