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10.1: Translations of Geometric Shapes

Difficulty Level: At Grade Created by: CK-12

Objectives

The lesson objectives for Translations of Geometric Shapes are:

  • What is a translation?
  • Graphing an image that has undergone a translation
  • Writing a rule for a translation
  • The properties of a translation

What is a Translation?

Concept Content

As you learned in the introduction, you are about to begin your study of transformations in geometry. A transformation is an operation that is created from some original image. There are four types of transformations and in this first lesson you will learn about translations. The other three transformations are reflections, dilations and rotations. Translations are made on an original image by moving every point on that image the same distance and in the same direction. If you look at the diagram below, you can see that the square ABCD is moved 10 units to the right. All points (ABCD) of the square have been moved 10 units to make the translated image (A^\prime B^\prime C^\prime D^\prime). The points have all been moved across the y-axis in the positive direction. The translation is therefore described as moving 10 units to the right. The original image (ABCD) is called a preimage. The final figure is called the image.

Translations, since the image will stay the same size and shape, involve moving the image around the Cartesian plane. Translations are often referred to as slides. If you look at the diagram below, it does look like the square ABCD slid across the y-axis to land in such as way that it produced the image A^\prime B^\prime C^\prime D^\prime.

In this concept you will learn to describe translations and in the process learn what they are. You will also learn to graph translations, write rules to describe them, and finally learn some of the properties of translations.

Guidance

Karen looked at the image below and stated that the image was translated thirteen units backwards. Is she correct? Explain.

Karen is somewhat correct in that the translation is moving to the left (backwards). The proper way to describe the translation is to say that the image STUV has moved 13 units to the left and 2 units up.

Examples

Example A

Describe the translation of the purple pentagon in the diagram below.

The pentagon is translated down 5 and over 7 to the right.

Example B

Describe the translation of the light blue triangle in the diagram to the right.

The blue triangle moves up 3 units and over 2 units to the left to make the green triangle image.

Example C

Describe the translation in the diagram below.

The original shape is translated down 2 and over 7 to the left.

Vocabulary

Image
In transformations, the final figure is called the image.
Preimage
In transformations, the original image (ABCD) is called a preimage.
Transformation
A transformation is an operation that is created from some original image. There are four types of transformations: translations, reflections, dilations and rotations.

Translations

Translations are made on an original image by moving every point on that image the same distance and in the same direction.

Guided Practice

1. Describe the translation of the pink triangle in the diagram below.

2. Describe the translation of the purple polygon in the diagram below.

3. Describe the translation of the blue hexagon in the diagram below.

Answers

1. The pink triangle is translated down 4 and over 2 to the left.

2. The purple polygon is translated up 2 and over 12 to the right.

3. The blue hexagon is translated down 2 and over 7 to the left.

Summary

In this first concept of Chapter Is it a Slide, a Flip, or a Turn? you were introduced to one of the four types of transformations, translations. Translations involve sliding an image on the Cartesian plane to the left, to the right, up and down. You describe the translations in this manner.

Problem Set

Describe the translation of the purple original figures in the diagrams:

Use the diagram below to describe the following translations:

  1. A onto B
  2. A onto C
  3. A onto D
  4. A onto E
  5. A onto F

Graphing an Image that has undergone a Translation

Concept Content

In this second concept of lesson Translations of Geometric Shapes, you will learn how to graph an image that has undergone a translation. Remember that a translation occurs when a preimage is slid across the Cartesian plane to form a new image. Each point in the preimage is moved the same distance and the same direction in the slide.

In the last lesson you moved the image up and down the y-axis and left or right on the x-axis. In this lesson you will create the images and preimages by first graphing them on a Cartesian plane. When graphing the translated image, it is often helpful to remember that:

  • translating up means add the translated unit to the y-coordinate of the (x, y) points in the preimage
  • translating down means subtract the translated unit from the y-coordinate of the (x, y) points in the preimage
  • translating right means add the translated unit to the x-coordinate of the (x, y) points in the preimage
  • translating left means subtract the translated unit from the x-coordinate of the (x, y) points in the preimage

Guidance

Triangle ABC has coordinates A(1, 1), B(8, 1) and C(5, 8). Draw the triangle on the Cartesian plane. Translate the triangle up 2 units and over 4 units to the right. State the coordinates of the resulting image. How far has each point moved?

The coordinates of the new image are A^\prime(5, 3), B^\prime(12, 3) and C^\prime(9, 10). The points have moved a distance of 4 units to the right and up 2 units.

Examples

Example A

Line \overline{A B} drawn from (–4, 2) to (3, 2) has been translated 3 units down and 7 units to the left. Draw the preimage and image and properly label each.

Example B

Triangle A is translated 3 units up and 5 units to the right to make triangle B. Find the coordinates of triangle B. On the diagram, draw and label triangle B.

Example C

The following figure is translated 4 units down and 6 units to the left to make a translated image. Find the coordinates of the translated image. On the diagram, draw and label the image.

Vocabulary

Image
In transformations, the final figure is called the image.
Preimage
In transformations, the original image (ABCD) is called a preimage.
Transformation
A transformation is an operation that is created from some original image. There are four types of transformations: translations, reflections, dilations and rotations.
Translations
Translations are made on an original image by moving every point on that image the same distance and in the same direction.

Guided Practice

1. Line \overline{S T} drawn from (–3, –3) to (–3, 8) has been translated 5 units up and 3 units to the right. Draw the preimage and image and properly label each.

2. The polygon below has been translated 4 units down and 10 units to the right. Draw the translated image and properly label each.

3. The gray pentagon is translated 5 units up and 8 units to the right to make a translated purple pentagon. Find the coordinates of the purple pentagon. On the diagram, draw and label the translated pentagon.

Answers

1.

2.

3.

Summary

In this first concept of Chapter Is it a Slide, a Flip, or a Turn? you were introduced to one of the four types of transformations, translations. You also learned how to describe the translations in this manner. In this lesson you worked from the translation description to graphing the translated images. You learned that:

  • translating up means add the translated unit to the y-coordinate of the (x, y) points in the preimage
  • translating down means subtract the translated unit from the y-coordinate of the (x, y) points in the preimage
  • translating right means add the translated unit to the x-coordinate of the (x, y) points in the preimage
  • translating left means subtract the translated unit from the x-coordinate of the (x, y) points in the preimage

Problem Set

  1. Translate figure MATH on the grid by moving it up 2 and over 5 to the right.
  2. Translate figure DEFGH on the grid by moving it down 3 and over 8 to the left.
  3. Translate figure LMO on the grid by moving it down 4 and over 3 to the right.
  4. Translate figure WXY on the grid by moving it up 5 and over 5 to the left.
  5. Translate the red figure on the grid by moving it up 6 and over 4 to the right.

For each of the following diagrams translate the preimage 3 units up and 4 units to the left to make the translated images. Find the coordinates of the preimages. On the diagram, draw and label the translated image.

For each of the following diagrams translate the preimage 7 units down and 3 units to the right to make the translated images. Find the coordinates of the preimages. On the diagram, draw and label the translated image.

Writing a Rule for a Translation

Concept Content

In this third concept with respect to translations, you will learn how to describe a translation using a rule. In the first lesson you used words to describe a translation. For example, you described translating a preimage to a new position by stating that the preimage moved up 3 and over 5 to the left. This is one way to describe the translation.

You could have used a notation to describe the translation. A notation such as T_{(3, \  5)}. This notation tells you to add 3 to the x values and add 5 to the y values. Sometimes this notation is written as T_{3, \ 5}(x, y) = (x+3, y+5). Another way to describe a translation using a rule is to write the translation as a mapping rule. A mapping rule has the following form (x, y) \rightarrow (x-7, y+5) and tells you that the x and y coordinates are translated to x - 7 and y + 5.The mapping rule is the most common method.

As with the other sections, when the translation is positive, you are adding. A positive value means the preimage has moved up or to the right. When the translation is negative, you are subtracting. A negative value means the preimage has moved down or to the left. In this concept you will write rules for translations as well as draw graphs from these rules.

Guidance

The figure below shows a pattern of a floor tile. Write the mapping rule for the translation of the two blue floor tiles.

To answer this question, let’s label one of the blue tiles as the preimage and another as the translated image. This way you can write the rule as to how the hexagons are being translated. While doing this, label one point and mark the coordinates so you can see how the point (and therefore the hexagon) is being translated.

The rule for writing the translation is:

p(x, y) \rightarrow p^\prime(x+a, y+b)

where “a” is the number of units the image is translated left or right and “b” is the number of units translated up or down and “p” is the point.

So you know that:

A: (-7, 2) \ \ A^\prime: (-2, 4)

A(x, y) \rightarrow A^\prime(x+a, y+b)

So: A(-7, 2) \rightarrow A^\prime(-7+a, 2+b) or A(-7, 2) \rightarrow A^\prime(-2, 4)

Therefore: -7+a&=-2 \quad \ \ and \quad \ \ 2+b=4 \\a&=5 \qquad \qquad \qquad \quad \ b=2

You can now write the general mapping rule for the translation of any point on the preimage to the translated image.

(x, y) \rightarrow (x+5, y+2)

Examples

Example A

Sarah describes a translation as point P moving from P(-2, 2) to P^\prime(1, -1). Write the mapping rule to describe this translation for Sarah.

P: (-2, 2) \ P^\prime: (1, -1)

P(x, y) \rightarrow P^\prime(x+a, y+b)

So: P(-2, 2) \rightarrow P^\prime(-2+a, 2+b) or P(-2, 2) \rightarrow P^\prime(1, -1)

Therefore: -2+a&=1 \quad \ \ and \quad \ \ 2+b=-1 \\a&=3 \qquad \qquad \qquad \ \ b=-3

You can now write the general mapping rule for the translation of any point on the preimage to the translated image.

(x, y) \rightarrow (x+3, y-3)

Example B

Mikah describes a translation as point D in a diagram moving from D(1, -5) to D^\prime(-3, 1). Write the mapping rule to describe this translation for Mikah.

D: (1, -5) \ D^\prime: (-3, 1)

D(x, y) \rightarrow D^\prime(x+a, y+b)

So: D(1, -5) \rightarrow D^\prime(1+a, -5+b) or D(1, -5) \rightarrow D^\prime(-3, 1)

Therefore: 1+a&=-3 \quad \ \ and \quad \ \ -5+b=1 \\a&=-4 \qquad \qquad \qquad \qquad b=6

You can now write the general mapping rule for the translation of any point on the preimage to the translated image.

(x, y) \rightarrow (x-4, y+6)

Example C

Write the mapping rule that represents the translation of the preimage A to the translated image J in the diagram below.

First, pick a point in the diagram to use to see how it is translated.

D: (-1, 4) \ D^\prime: (6, 1)

D(x, y) \rightarrow D^\prime(x+a, y+b)

So: D(-1, 4) \rightarrow D^\prime(-1+a, 4+b) or D(-1, 4) \rightarrow D^\prime(6, 1)

Therefore: -1+a&=6 \quad \ \ and \quad \ \ 4+b=1 \\a&=7 \qquad \qquad \qquad \ \ b=-3

You can now write the general mapping rule for the translation of any point on the preimage to the translated image.

(x, y) \rightarrow (x+7, y-3)

Vocabulary

Mapping Rule
A mapping rule has the following form (x, y) \rightarrow (x-7, y+5) and tells you that the x and y coordinates are translated to x-7 and y + 5.
Translations
Translations are made on an original image by moving every point on that image the same distance and in the same direction.

Guided Practice

1. Jack describes a translation as point J moving from J(-2, 6) to J^\prime(4, 9). Write the mapping rule to describe this translation for Jack.

2. Write the mapping rule that represents the translation of the red triangle to the translated green triangle in the diagram below.

3. The following pattern is part of wallpaper found in a hotel lobby. Write the mapping rule that represents the translation of one blue trapezoid to a translated blue trapezoid shown in the diagram below.

Answers

1. J: (-2, 6) \quad J^\prime: (4, 9)

J(x, y) \rightarrow J^\prime(x+a, y+b)

So: J(-2, 6) \rightarrow J^\prime(-2+a, 6+b) or J(-2, 6) \rightarrow J^\prime(4, 9)

Therefore: -2+a&=4 \quad \ \ and \quad \ \ 6+b=9 \\a&=6 \qquad \qquad \qquad \ \ b=3

You can now write the general mapping rule for the translation of any point on the preimage to the translated image.

(x, y) \rightarrow (x+6, y+3)

2. In order to write a mapping rule, choose one point on the preimage (the red triangle) and then the translated point on the green triangle to see how the point has moved.

Once you have the two points identified, work to find the mapping rule.

T: (8, 5) \quad T^\prime: (5, 0)

T(x, y) \rightarrow T^\prime(x+a, y+b)

So: T(8, 5) \rightarrow T^\prime(8+a, 5+b) or T(8, 5) \rightarrow T^\prime(5, 0)

Therefore: 8+a&=5 \quad \ \ and \quad \ \ 5+b=0 \\a&=-3 \qquad \qquad \quad \ \ \ b=-5

You can now write the general mapping rule for the translation of any point on the preimage to the translated image.

(x, y) \rightarrow (x-3, y-5)

3. If you look closely at the diagram below, there two pairs of trapezoids that are translations of each other. Therefore you can choose one blue trapezoid that is a translation of the other and pick a point to find out how much the shape has moved to get to the translated position.

D: (-10, 2) \quad D^\prime: (-6, -3)

D(x, y) \rightarrow D^\prime(x+a, y+b)

So: D(-10, 2) \rightarrow D^\prime(-10+a, 2+b) or D(-10, 2) \rightarrow D^\prime(-6, -3)

Therefore: -10+a&=-6 \quad \ \ and \quad \ \ 2+b=-3 \\a&=4 \qquad \qquad \qquad \quad \ \ b=-5

You can now write the general mapping rule for the translation of any point on the preimage to the translated image.

(x, y) \rightarrow (x+4, y-5)

Summary

In this last concept of lesson Translations of Geometric Shapes you have learned the final step in working with translations, that is, to write the mapping rule to describe the translation. The mapping rule allows you to quickly determine how much each point has moved both on the x-axis and on the y-axis to translate the preimage to the translated image. The most common form of writing a rule for a translation is (x, y) \rightarrow (x+a, y+b) where “a” represents the number of units the translated point has moved left or right on the x-axis and “b” represents the number of units the point has moved up or down on the y-axis.

There are other forms of writing a mapping rule that you may see in math literature. In the last examples you wrote mapping rules like:

(x, y) \rightarrow (x-4, y+5)

You could have indicated the translation by writing:

T_{(-4, 5)} (x, y) = (x-4, y+5)

where T_{(-4, 5)} represents the points moving left 4 units and up 5 units. This notation is less common but does exist in some mathematical texts.

Problem Set

Write the mapping rule to describe the movement of the points in each of the translations below.

  1. S(1, 5) \rightarrow S^\prime(2, 7)
  2. W(-5, -1) \rightarrow W^\prime (-3, 1)
  3. Q(2, -5) \rightarrow Q^\prime(-6, 3)
  4. M(4, 3) \rightarrow M^\prime(-2, 9)
  5. B(-4, -2) \rightarrow B^\prime(2, -2)

Write the mapping rule that represents the translation of the preimage A to the translated images in the diagrams below.

The Properties of a Translation

Concept Content

As you learned earlier, translations involve sliding all points on a preimage the same distance and in the same direction. Translations do not involve rotating, changing the size, or anything else. Translations are just moving.

In this last concept you will look at the properties of translations. Since the translated image is congruent to the preimage, translations are considered to have the following properties:

  • line segments are the same length (distance)
  • angle measures in a figure remain the same (angles)
  • lettering order remains the same (orientation)
  • parallel lines remain parallel (same slopes)
  • points that are on each line remain on the line for the translated image (collinear)

Guidance

The triangle ABC is drawn such that the vertices are at A(1, 1), B(5, 5) and C(-2, 4). Triangle DEF is drawn such that the vertices are at D(-6, -3), E(-2, 1), and F(9, 0).

a) Measure each angle either with a protractor or using geometry software. What angles are congruent?

b) Measure each side length. Are the distances the same?

c) Can you conclude that the triangle DEF is a translation of triangle ABC?

a) The measures of each angle are shown in the diagram.

b) Congruent angles have the same measure. Since \angle A = \angle D, \angle B = \angle E, \angle C = \angle E, the angles are congruent.

The side lengths are also the same (m \overline{A B} = m \overline{D E}, m \overline{A C} = m \overline{D F}, and m \overline{B C} = m \overline{E F}).

c) Since the angle measures are the same and the lengths of each side are the same, the triangles are congruent. Each point it moved down 7 units and over 4 units to the left. With all of this, you can conclude that triangle DEF is a translated image of the preimage triangle ABC.

Examples

Example A

Graph the following two squares to determine if Square X is translated to form Square Y.

Square X Vertices: A(2, 3), B(8, 3), C(8, 8), and D(2, 8).

Square Y Vertices: W(-3, -4), X(3, -4), Y(3, 1), and Z(-3, 1).

The side lengths are also the same (m \overline{A B} = m \overline{W X}, m \overline{BC} = m \overline{XY}, m \overline{C D} = m \overline{Y Z}, and m \overline{A D} = m \overline{W Z}).

Since the side measures are the same (both shapes are squares and therefore all angles are 90^\circ), Square X is congruent to Square Y. Each point it moved down 7 units and over 5 units to the left. With all of this, you can conclude that Square Y is a translated image of the preimage Square X.

Example B

Describe how you would know if image T is translated onto image T^\prime.

As indicated on the graph, the side measures are the same and the angle measures are all the same. Image T is congruent to Image T^\prime. Each point it moved down 5 units and over 5 units to the right. With all of this, you can conclude that Image T^\prime is a translated image of the preimage Image T.

Example C

Describe how you would know if image A is translated onto image B.

As indicated on the graph, the side measures are the same and the angle measures are all the same. Image A is congruent to Image B. Each point it moved up 1 unit and over 10 units to the right. With all of this, you can conclude that Image B is a translated image of the preimage Image A.

Vocabulary

Congruent Angles
Congruent angles have the same measure.
Translations
Translations are made on an original image by moving every point on that image the same distance and in the same direction.

Guided Practice

1. Describe how you would know if image A is translated onto image B.

2. Describe how you would know if image S is translated onto image T.

3. Is ABCDEF a translated image of A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime? How do you know?

Answers

1.

As indicated on the graph, the side measures are the same and the angle measures are all the same. Image A is congruent to Image B. Each point it moved down 3 units and over 5 units to the left. With all of this, you can conclude that Image B is a translated image of the preimage Image A.

2.

As indicated on the graph, the side measures are the same and the angle measures are all the same. Image S is congruent to Image T. Each point it moved up 6 units only. With all of this, you can conclude that Image T is a translated image of the preimage Image S.

3.

As indicated on the graph, the side measures are the same and the angle measures are all the same. Image ABCDEF is congruent to Image A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime. Each point it moved down 1 unit and over 7 units to the right. With all of this, you can conclude that Image A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime is a translated image of the preimage Image ABCDEF.

Summary

The previous three lessons you worked with translations, a transformation of a figure that involves moving the figure along the Cartesian plane. In this lesson you looked more closely at the properties of transformations. These properties include:

  • line segments are the same length (distance)
  • angle measures in a figure remain the same (angles)
  • lettering order remains the same (orientation)
  • parallel lines remain parallel (same slopes)
  • points that are on each line remain on the line for the translated image (collinear)

Through the use of geometry software, it is easy to measure both the angles and the side lengths to determine if they are congruent, thus satisfying the first two properties. If points are each moving in the same direction and at the distance measure, you can be sure that the preimage is translated to form the new image.

Problem Set

Describe how you would know if image A is translated onto image B in each of the following diagrams.

In each case, is the second image a translation of preimage A? How do you know?

Summary

In this first lesson of chapter Is it a Slide, a Flip, or a Turn? you have been introduced to translations. Remember that translations are just one of the four different transformations that can be done to a shape. There are also rotations, reflections, and dilations. Translations are simply movements. You can move a figure or a shape up and down on the y-axis. You can also move the shape or figure left or right on the x-axis. Or you can do both. The original image is known as the preimage and when you move the preimage to form a translated image, you have to describe the translation.

You first learned to describe the translation using words. So, you learned when the translation is positive, you are adding. A positive value means the preimage has moved up or to the right. When the translation is negative, you are subtracting. A negative value means the preimage has moved down or to the left. You ended this lesson learning how to write a mapping rule to describe a translation. The most common method for writing a mapping rule was using mapping notation (x, y) \rightarrow (x+a, y+b) where “a” represents how many units your preimage has moved left or right on the x-axis and “b” represents the number of units your preimage has moved up or down on the y-axis. So, for example, you may see the mapping rule (x, y) \rightarrow (x+4, y-5) which describes a translation of a preimage moved 4 units to the right and 5 units down. If you were to graph this, you would see the following:

Finally in this lesson, you learned that translations have the following properties:

  • line segments are the same length (distance)
  • angle measures in a figure remain the same (angles)
  • lettering order remains the same (orientation)
  • parallel lines remain parallel (same slopes)
  • points that are on each line remain on the line for the translated image (collinear)

You used these properties to prove that one image is a translation of another image by finding the lengths of the preimage and the translated image as well as the angle measures in both images. You then found where the points moved from the preimage to the translated image. All of these properties combined proved that the one image was a translation of the other.

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May 28, 2014

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Jan 14, 2015
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