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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.*
### Catch-Up and Review

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

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

Explore

In the following applet, arrow $v$ dictates how $△ABC$ will move to get $△A_{′}B_{′}C_{′}.$ Try changing the direction and length of the arrow to see different types of movements.

Notice how $△ABC$ and $△A_{′}B_{′}C_{′}$ are congruent and how the orientation of $△A_{′}B_{′}C_{′}$ matches the orientation of $△ABC.$ Furthermore, the segments $AA_{′},$ $BB_{′},$ and $CC_{′}$ are all parallel to arrow $v,$ and all four segments have the same length.

Discussion

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.

Discussion

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 $a$ is added to the $x-$coordinate of every point of the preimage and a value $b$ is added to the $y-$coordinate of every point of the preimage.

$(x,y)→(x+a,y+b) $

Positive $a$ values indicate a translation to the right and positive $b$ values correspond to a translation up.
Conversely, negative $a$ values indicate a translation to the left and negative $b$ values correspond to a translation down.

Translations preserve the side lengths and angle measures of the involved geometric objects.

Example

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 $P$ 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, $P_{1}$ and $P_{2}.$

a Let $P$ be the preimage of the pool before a transformation. Which figure represents the image of $P$ after a pure translation?

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b The translation can be written as $(x,y)→(x+a,y+b).$ Use the grid on the sketch to find the $a$ and $b$ values of the translation of the pool.

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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.

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

A pure translation takes a preimage $P$ and makes it into an image that is congruent and has the same orientation as $P.$ In the sketch, both $P_{1}$ and $P_{2}$ are congruent to $P,$ so all that remains is to determine which image has the same orientation as $P.$ Notice that the longest side of $P$ is horizontal in the sketch.

This preimage will now be compared to images $P_{1}$ and $P_{2}.$ The longest side of image $P_{1}$ is vertical in the sketch, so it does not have the same orientation as $P.$ This means that $P_{1}$ is **not** a pure translation of $P.$ On the other hand, $P_{2}$ matches $P$ exactly.

Since $P_{2}$ is congruent to $P$ and has the same orientation, it is a pure translation of $P.$

b Given the coordinates $(x,y)$ of each point in a preimage, a translation of the point can be written in the following form.

$(x,y)→(x+a,y+b) $

In this case, it is not necessary to know the coordinates $(x,y),$ only the values $a$ and $b$ that are relevant to the translation. It was found in Part A that $P_{2}$ is the image of Zain's pool after a translation, so the focus will be in this image.
The $a$ value of the translation corresponds to the horizontal displacement of the translation, while the $b$ 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 $a$ and $b.$ Just keep in mind that they have to be matching corners!

The horizontal displacement is $4$ units to the right, so $a=4.$ Likewise, the vertical displacement is $7$ units up, so $b=7.$

Pop Quiz

In the following applet, $△ABC$ is translated to map onto $△A_{′}B_{′}C_{′}.$ Write the $a$ and $b$ values of the translation.

Example

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 $7$ units to the left and $1$ unit up. Find the coordinates of the pot after the translation.{"type":"text","form":{"type":"point2d","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"decimal":false,"function":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text1":"-3","text2":"-1"}}

Find the initial coordinates of the pot. To translate it $7$ units to the left and $1$ unit up, subtract $7$ from the $x-$coordinate and add $1$ to the $y-$coordinate.

Zain's mom wants the builders to return the planter pot back to where it was, which they can do by translating it $7$ units to the left and $1$ unit up. Begin by finding the current coordinates of the pot.

The horizontal part of the translation is $7$ units to the $left.$ This means to $subtract$ $7$ from the current $x-$coordinate of the pot.$4−7=-3 $

On the other hand, the vertical part of the translation is $1$ unit $up,$ so $add$ $1$ to the current $y-$coordinate of the pot.
$-2+1=-1 $

The $x-$coordinate of the pot after the translation is $-3$ and its $y-$coordinate is $-1.$
$(4−7,-2+1)=(-3,-1) $

The following applet can be used to visualize the translation.
Explore

In the following applet, the vertices of $△ABC$ can be moved. The slider bar adjusts the slope of the line $ℓ.$ Once everything is set, reflect $△ABC$ across line $ℓ.$

Is there any relationship between $AA_{′}$ and $ℓ?$ If so, do $BB_{′}$ and $ℓ$ have the same relationship? What about $CC_{′}$ and $ℓ?$

Discussion

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.

More precisely, a reflection across a line $ℓ$ maps every point $A$ in the plane onto its image $A_{′}$ such that one of the following statements is satisfied.

- If $A$ is on the line $ℓ,$ then $A$ and $A_{′}$ are the same point.
- If $A$ is not on the line $ℓ,$ then $ℓ$ is the perpendicular bisector of $AA_{′}.$

Example

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

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 $CC_{′}.$

Next, construct the perpendicular bisector of $CC_{′}.$ 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

Drag the points in the following applet to draw the line of reflection used to map $△ABC$ onto $△A_{′}B_{′}C_{′}.$ 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.
### 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 $180_{∘}$ 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.