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There are lots of things that can be done to a given a geometric figure to obtain a new one. Since a figure is transformed into a new one, these are called transformations.
A quadrilateral ABCD and its image A'B'C'D' under a transformation

Transformations are useful to describe the relationship between two figures. This lesson will discuss two particular types of transformations: translations and reflections.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Moving a Triangle

In the following applet, arrow dictates how will move to get Try changing the direction and length of the arrow to see different types of movements.
Notice how and are congruent and how the orientation of matches the orientation of Furthermore, the segments and are all parallel to arrow and all four segments have the same length.
Discussion

Transformations of Geometric Objects

A transformation is a function that changes a figure in a particular way — it can change the position, size, or orientation of a figure. The original figure is called the preimage and the figure produced is called the image of the transformation. A prime symbol is often added to the label of a transformed point to denote that it is an image.

A quadrilateral ABCD and its image A'B'C'D' under a transformation
Discussion

Translations of Geometric Objects

A translation is a transformation that moves every point of a figure the same distance in the same direction. To find the coordinates of a geometric object after a translation, a value is added to the coordinate of every point of the preimage and a value is added to the coordinate of every point of the preimage.
Positive values indicate a translation to the right and positive values correspond to a translation up.
Conversely, negative values indicate a translation to the left and negative values correspond to a translation down.
Translations preserve the side lengths and angle measures of the involved geometric objects.
Example

Placing a Pool in the Backyard

Zain's parents want to renovate their backyard. The lot is pretty wide and, once the weeds are taken care of, it could be used for plenty of things. The first thing that comes to Zain's mind is to put in a pool, so they make a sketch of the backyard to show their idea to their parents. They use the letter to represent the pool.

Zain's dad says that the pool should not be right against the gate to the backyard, so he suggests two possible alternatives, and

a Let be the preimage of the pool before a transformation. Which figure represents the image of after a pure translation?
b The translation can be written as Use the grid on the sketch to find the and values of the translation of the pool.

Hint
a A translation maps figures into congruent figures that have the same orientation.
b Focus on any corner of the pool. Count how many squares it is moved vertically and horizontally.

Solution
a Consider the pool location suggestions made by Zain's dad.

A pure translation takes a preimage and makes it into an image that is congruent and has the same orientation as In the sketch, both and are congruent to so all that remains is to determine which image has the same orientation as Notice that the longest side of is horizontal in the sketch.

This preimage will now be compared to images and The longest side of image is vertical in the sketch, so it does not have the same orientation as This means that is not a pure translation of On the other hand, matches exactly.

Since is congruent to and has the same orientation, it is a pure translation of

b Given the coordinates of each point in a preimage, a translation of the point can be written in the following form.
In this case, it is not necessary to know the coordinates only the values and that are relevant to the translation. It was found in Part A that is the image of Zain's pool after a translation, so the focus will be in this image.

The value of the translation corresponds to the horizontal displacement of the translation, while the value is the vertical displacement. Since every point of the preimage is translated in the same way, any particular corner of the pool can be used along with the grid to find and Just keep in mind that they have to be matching corners!

The horizontal displacement is units to the right, so Likewise, the vertical displacement is units up, so

Pop Quiz

Identifying Translations

In the following applet, is translated to map onto Write the and values of the translation.

Performing random translations to random triangles
Example

Replacing the Planter

The renovations are now taking place, so there is a lot going on in the backyard. During the morning, Zain's mom goes out to water some plants that were on a planter pot when she noticed that they are in the wrong place.

The planter was probably in the way when the builders were working on the pool. However, the plants need to be in the correct spot to get just the right amount of sunlight, so she kindly requests the workers to put them back where they belong when they are finished for the day. To find the original position of the pots, Zain placed a coordinate plane on the backyard plans.

Zain's mom asks the builders to translate the pot units to the left and unit up. Find the coordinates of the pot after the translation.

Hint

Find the initial coordinates of the pot. To translate it units to the left and unit up, subtract from the coordinate and add to the coordinate.

Solution

Zain's mom wants the builders to return the planter pot back to where it was, which they can do by translating it units to the left and unit up. Begin by finding the current coordinates of the pot.

The horizontal part of the translation is units to the This means to from the current coordinate of the pot.
On the other hand, the vertical part of the translation is unit so to the current coordinate of the pot.
The coordinate of the pot after the translation is and its coordinate is
The following applet can be used to visualize the translation.
Explore

Reflecting a Triangle

In the following applet, the vertices of can be moved. The slider bar adjusts the slope of the line Once everything is set, reflect across line
Is there any relationship between and If so, do and have the same relationship? What about and
Discussion

Reflections of Geometric Objects

A reflection is a transformation in which every point of a figure is reflected across a line. The line across the points are reflected is called the line of reflection and acts like a mirror.
Triangle being reflected across a movable line
More precisely, a reflection across a line maps every point in the plane onto its image such that one of the following statements is satisfied.
  • If is on the line then and are the same point.
  • If is not on the line then is the perpendicular bisector of
Segment AA' intersects line ell perpendicularly, and line ell bisects segment AA'. Points B and B' coincide.
 Like translations, reflections also preserve side lengths and angle measures. However, reflections can change the orientation of the preimage.
Example

Reflections In the Pool Tiles

Once the pool was installed, Zain went to take a look at it. They noticed a unique pattern printed in the pool tiles.

Zain thinks of this as a transformation and writes the following labels for the vertices of both figures.

It seems that there is some sort of reflection between the tiles, but Zain is not completely sure. If there was a reflection, it would be possible to find a line of reflection. Help Zain find the line of reflection.

Answer

Hint

Draw segments that connect every point of the preimage with its corresponding point in the image — for example, The line of reflection is the perpendicular bisector of any segment connecting a point to its image.

Solution

When a point is reflected across a line, its image is such that the line of reflection is the perpendicular bisector of the segment that connects the point to its image. To find the line of reflection of the pool tiles, start by drawing a segment that connects a vertex and its image. For instance, draw

Next, construct the perpendicular bisector of This will represent the line of reflection used to make the pattern in the pool tiles.

With this, the line of reflection can be finally drawn.

Pop Quiz

Drawing Lines of Reflection

Drag the points in the following applet to draw the line of reflection used to map onto To do so, place the two points so they lie on the line of reflection. The measuring tool can be used to find the midpoint between corresponding vertices of the image and preimage, and they can also be used to find right angles.
Performing random reflections to random triangles

Extra

 How to Use the Measuring Tool

The measuring tool is useful for finding the midpoint between a point and its image. Place the ends of the measuring tool on a point and its corresponding image.

Next, move the middle point so that the angle of the measuring tool is and the length of both segments is equal. This will make sure the middle point lies on the line of reflection.

Repeat this process on another pair of corresponding vertices to find a second point that lies on the line of reflection.

Now that two points that lie on the line of reflection have been identified, that line can be drawn.

Pop Quiz

Reflecting Triangles

Place points and where they should be to reflect across the given line. The measuring tool can be used to find segments perpendicular to the line of reflection.

Performing random reflections to random triangles
Discussion

Reflections in the Coordinate Plane

In the coordinate plane, there is a particular relationship between the coordinates of a point and those of its image after a reflection across the coordinate axes. Investigate each relationship by using the following applet.
Applet to investigate the coordinates of a point after a reflection across the coordinate axes and the lines y=x and y=-x
Drawn from diagram, the following relations can be determined.
  • The image of after a reflection across the axis is
  • The image of after a reflection across the axis is
Example

Placing Chairs in the Backyard

It is always a good idea to have a picnic table close to the pool for snacks and drinks and to hang out while drying off, so Zain added this to the sketch, along with two chairs.

Zain noted that the table is placed along the axis they drew when trying to figure out where to return the planter pot.

To keep things symmetric, Zain wants to add two more chairs so that they are reflections across the axis of the other two chairs. Write the coordinates of the image of the left chair after a reflection across the axis.

Hint

Identify the coordinates of the left chair. A reflection across the axis can be plotted by changing the sign of the coordinate of the preimage.

Solution

Two more chairs are to be placed so that they are reflections across the axis of the currently shown chairs.

Start by finding the coordinates of the left chair.

To find the reflection of an object across the axis, just change the sign of its coordinate.
Now that the coordinates of the image have been identified, Zain can place the chairs!

The right chair can also be reflected by following the same procedure.

Closure

A Brief Talk About Glide Reflections

This lesson introduced two types of transformations of geometric objects. Translations move every point of the preimage the same distance in the same direction.
Reflections move every point in the preimage over to the line of reflection, which acts like a mirror.
These transformations can be combined to make a new transformation. When the figure is translated in a direction parallel to the line of reflection this is called a glide reflection. In the following applet, is translated in the direction and length of arrow and then is reflected across the line
Performing a Glide Reflection on a Triangle
A glide reflection can also be done by performing the reflection before the translation.
Performing a Glide Reflection on a Triangle
Note that the image is the same in both cases, which means that, in the case of a glide reflection, the image does not depend on the order of the transformations.
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