Topics
5. Rigid Transformations
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Geometry

5.01 Translations

Lesson
Practice

Translations

A transformation of a figure is a mapping that changes the figure's size or position in space. We can also think of a transformation as a function, where the input values are the points that make up the figure that is being transformed.

The figure before it is transformed is called the preimage. The figure after it has been transformed is called the image. The points that make up the image would be considered the outputs of the function transformation.

Exploration

Drag the points on the preimage to change its shape. The image is created by translating the preimage. Drag the sliders to translate the image horizontally and vertically.

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  1. If you are given the coordinates of the image, what information would you need to find the coordinates of the preimage and vice versa?

  2. What can we conclude about the corresponding segment between any two points on the preimage and its image?

A rigid transformation (or isometry) is a transformation that does not change the size or shape of a figure.

Translation

A translation is a rigid transformation in which an image is formed by moving every point on the preimage the same distance in the same direction.

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It is common to label the vertices of figures with letters and to use an apostrophe, called a prime, to label vertices of the transformed image. For example, if A was the preimage, then A' (spoken as "A-prime") is the image.

We can describe a translation algebraically using coordinates:

  • Coordinate notation: The translation \left(x,y\right) \to \left(x+h,y+k\right) takes the preimage and moves it h units horizontally, and k units vertically to obtain the image.

The movement h units horizontal and k units vertical can be represented as a directed line segment. So we can think of a translation as moving the preimage along the directed line segment to get the image.

Since every point in the preimage is moved in the same direction and distance, every line segment from a preimage point to its corresponding image point will be parallel to the directed line segment that represents the translation.

An image of a polygon and its pre image. Rays are drawn from each vertex of the pre image to its correspnding image point. The rays are parallel.

Examples

Example 1

For the following graph:

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Describe the transformation in words.

Worked Solution
Create a strategy

We need to identify the direction the preimage has been moved and the distance each point has moved to obtain the image.

Apply the idea

The preimage has been moved four units to the left and two units down to obtain the image.

Reflect and check

It can be easier to work out the description by looking at one vertex and seeing how that moved. Then we can confirm by trying the same translation with the other vertices, or even other points on the sides of the triangle.

Example 2

Triangle ABC is to be translated 4 units right and 2 units down.

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Write the transformation in coordinate form.

Worked Solution
Create a strategy

We want to translate the whole triangle 4 units right and 2 units down. We can write the transformation in coordinate form by seeing how these transformations would affect the x and y-values of the preimage.

Apply the idea

To translate the triangle to the right, we need to increase the x-values by adding 4 units to them. To translate the triangle down, we need to decrease the y-values by subtracting 2 units from them.

The transformation in coordinate form is:

\left(x,y\right) \to \left(x+4,y-2\right)

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Draw the image.

Worked Solution
Create a strategy

To draw the image of the translation, we can move each of the coordinates on the preimage right 4 units and down 2 units, then connect them to form the full image.

Alternatively, we can identify the coordinates of the preimage and find the coordinates of the image using the coordinate notation we found in part (a).

Apply the idea

Using the coordinate notation from part (a), we can substitute the coordinates of the preimage to find the coordinates of image:

A\left(-4,3 \right) becomes \left(-4+4,3-2\right) or A'\left(0,1\right).

B\left(-1,2 \right) becomes \left(-1+4,2-2\right) or B'\left(3,0\right).

C\left(-3,-1 \right) becomes \left(-3+4,-1-2\right) or C'\left(1,-3\right).

Now that we have identified the new coordinates, we can plot the image.

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Reflect and check

If we construct directed line segments from each vertex in the preimage to its corresponding vertex in the image, we can visually see that they are all parallel and are also the same length.

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Example 3

Draw the image given from the transformation \left(x,y\right) \to \left(x+1,\,y-4\right) on the preimage:

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Worked Solution
Create a strategy

The transformation tells us each value of x needs to be increased by 1 and each value of y needs to be decreased by 4. We can take each point on the figure and apply these transformations to them.

Apply the idea

Point \left(1,3 \right) becomes \left(1+1,3-4\right) or \left(2,-1\right).

Point \left(2,1 \right) becomes \left(2+1,1-4\right) or \left(3,-3\right).

Point \left(3,2 \right) becomes \left(3+1,2-4\right) or \left(4,-2\right).

Point \left(2,3 \right) becomes \left(2+1,3-4\right) or \left(3,-1\right).

Plotting these points on the graph, we obtain the image.

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Example 4

A triangle is translated 5 units to the right and 3 units down. The coordinates of the image after this translation are S'\left(7, 2\right),\, T'\left(4, 0\right) and U'\left(2, 3\right).

Determine the coordinates of the preimage points S,T, and U before the translation.

Worked Solution
Create a strategy

To find the preimage points before the translation, we can determine what operation was used to translate image, then apply the opposite operation to undo the translation.

The triangle was translated 5 units to the right, which means 5 was added to each x-coordinate. To go backwards to find preimage, we would then need to subtract 5 from every x-coordinate.

The triangle was also translated 3 units down, which means 3 was subtracted from every y-coordinate. To go backwards to find the preimage, we would then need to add 3 to every y-coordinate.

Apply the idea

For point S'\left(7, 2\right), the original coordinates of point S are found by subtracting 5 from the x-coordinate and adding 3 to the y-coordinate:

S'\left(7-5, 2+3\right) = S\left(2, 5\right)

Applying the same method to points T' and U', we get:

T'\left(4-5, 0+3\right) = T\left(-1, 3\right)

U'\left(2-5, 3+3\right) = U\left(-3, 6\right)

Therefore, the preimage points are S\left(2, 5\right),\, T\left(-1, 3\right), and U\left(-3, 6\right).

Reflect and check

We can graph the image and preimage to check our answer:

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Looking at the triangles, we can see that the preimage was tranlsated right 5 units and down 3 units.

We can also see that 5 was added to each x-coordinate and 3 was subtracted from each y-coordinate to go from the preimage to image.

Idea summary

A translation is a rigid transformation in which an image is formed by moving every point on the preimage the same distance in the same direction.

Using coordinate notation: the translation \left(x,y\right) \to \left(x+h,\,y+k\right) takes the preimage and moves it h units horizontally, and k units vertically to obtain the image.

Outcomes

G.RLT.3

The student will solve problems, including contextual problems, involving symmetry and transformation.

G.RLT.3ci

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: i) translations;

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