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Apart from rotations and translations, there is a third type of rigid motion called **reflections**. In a plane, reflections act similar to mirrors. However, unlike a mirror, a figure is reflected across a line. In this lesson, the formal definition and properties of reflections will be developed.
### Catch-Up and Review

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

In the following applet, the vertices of $△ABC$ can be moved. Also, the slope of the line $ℓ$ can be set by moving the slider point. 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 $ℓ?$

As can be checked in the previous exploration, after a point $A$ is reflected across a line $ℓ,$ the segment connecting $A$ with its image is perpendicular to $ℓ.$ Additionally, the line $ℓ$ intersects $AA_{′}$ at its midpoint.

From these two facts, it can be concluded that $ℓ$ is the perpendicular bisector of $AA_{′}.$ With this information in mind, reflections can be defined properly.

A reflection is a transformation in which every point of a figure is reflected in a line. The line in which 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 into 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 $AA_{′}.$

The principal of Jefferson High wants to build a physics lab by the chemistry lab. The plan, seen from the sky, is that the new building looks like a reflection of the chemistry lab through the walkway that connects the soccer field with the library.

Perform a reflection to the chemistry lab across the walkway in order to draw the physics lab.

To reflect the chemistry lab, reflect each corner of the building. The physics lab is the quadrilateral formed by the images. Remember that the image of a point that is on the line of reflection is the same point.

For simplicity, start by labeling each corner of the quadrilateral and the walkway.

To reflect $ABCD$ across $ℓ,$ a reflection can be performed on each vertex, one at a time. For example, to reflect $A$ is a good start. To do so, follow the definition of reflections. First, draw a line perpendicular to $ℓ$ passing through $A.$

Then $A_{′}$ can be plotted as the point on line $m$ where its distance to $ℓ$ is the same as the distance from $A$ to $ℓ.$

The same steps can be applied to reflect vertices $B,$ $C,$ and $D.$ Notice that because both $C$ and $D$ are on the line $ℓ,$ their images will maintain their same point locations, respectively.

Finally, the image of $ABCD$ under a reflection across the line $ℓ$ is the quadrilateral formed by $A_{′},$ $B_{′},$ $C_{′},$ and $D_{′}.$ This quadrilateral represents the physics lab.

Reflections can be performed by hand with the help of a straightedge and a compass.

To reflect $△ABC$ across the line $ℓ,$ follow these three steps.

1

Draw an Arc Centered at $A$ that Intersects $ℓ$ at Two Points

2

Draw Two Arcs Centered at the Intersection Points

With the same compass setting, place the compass tip on $P$ and draw an arc. Keeping the same setting, place the compass tip on $Q$ and draw a second arc. The intersection point of these two arcs is the image of $A.$

Notice that both arcs need to be drawn on the side of $ℓ$ not containing vertex $A.$

3

Repeat the Previous Steps for the Other Vertices

To reflect $B$ and $C,$ repeat the two previous steps.

The image of $△ABC$ after the reflection is the triangle formed by $A_{′},$ $B_{′},$ and $C_{′}.$

In the coordinate plane, there is a particular relationship between the coordinates of a point and those of its image after a reflection across certain lines worth considering. These lines are the coordinate axes and lines $y=x$ and $y=-x.$ Investigate each relationship by using the following applet.

Drawn from diagram, the following relations can be determined.

- The image of $(a,b)$ under a reflection across the $x-$axis is $(a,-b).$
- The image of $(a,b)$ under a reflection across the $y-$axis is $(-a,b).$
- The image of $(a,b)$ under a reflection across the line $y=x$ is $(b,a).$
- The image of $(a,b)$ under a reflection across the line $y=-x$ is $(-b,-a).$

Up to this point, how to perform a reflection, when already given the line of reflection, has been understood. Now, consider a case if given a figure and its image under a reflection. How can the line of reflection be found? This question is answered in the following example.

While visiting a museum, Tearrik saw a painting containing the word MATH

and two pentagons. The picture caught his attention. There is some sort of reflection but he want's to know for sure.

Tearrik analyzed the painting. He determined that the picture was made by performing a reflection. Show how Tearrik figured that out and draw the line of reflection used in the picture.

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 connecting the point to its image. Therefore, to find the line of reflection of the painting, start by drawing a segment that connects a vertex and its image. For instance, draw $CC_{′}.$

Next, construct the perpendicular bisector of $CC_{′}.$ That will represent the line of reflection used to make the painting.

Notice that drawing only one segment that connects a point to its image is enough to find the line of reflection. Before performing the reflection, the painting looked as follows.

In the following applet, there are two possible requests.

- Reflect $△ABC$ across the given line.
- Draw the line of reflection used to map $△ABC$ onto $△A_{′}B_{′}C_{′}.$

To reflect $△ABC,$ place points $A_{′},$ $B_{′},$ and $C_{′}$ where they should be after the reflection is applied. To draw the line of reflection, place the two points, so they lie on the line of reflection.

In previous lessons, the composition of rotations and translations were studied. Now it is time to learn about the composition of reflections. The first case to consider is when the lines of reflection are parallel.

Using a drone, Kevin took a photo of the roof of his house, Main Street, and Euclid Sreet — they are parallel streets.

Perform the following transformations to Kevin's house.

- Reflect Kevin's house across the midline of Main Street.
- Reflect the obtained image across the midline of Euclid Street.

Draw both reflections over the original photo. Is there a single transformation that maps Kevin's house onto the final image?

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To reflect Kevin's house across the midline of Main Street, start by drawing lines perpendicular to the line of reflection passing through each vertex of the house. Label each vertex and line for added clarity.

The image of $A$ after the reflection is a point $A_{′}$ on $ℓ_{2}$ such that $A_{′}$ and $A$ are equidistant from the midline. The images of $B,$ $C,$ and $D$ can be located in a similar way. By connecting $A_{′},$ $B_{′},$ $C_{′}$ and $D_{′},$ the image of Kevin's house after performing the first reflection can be obtained.

Applying a similar reasoning, the second reflection can be performed. After the second reflection is performed, the image of Kevin's house lies to the right of Euclid Street.

To determine whether there is a single transformation that maps Kevin's house to the final image, label the vertices of the initial polygon and its images.

From the previous diagram, the following conclusions can be drawn.

- Both $ABCD$ and $A_{′′}B_{′′}C_{′′}D_{′′}$ have the same orientation, in contrast to $ABCD$ and $A_{′}B_{′}C_{′}D_{′}.$
- The corresponding sides of $ABCD$ and $A_{′′}B_{′′}C_{′′}D_{′′}$ are parallel.
- The vectors $AA_{′′},$ $BB_{′′},$ $CC_{′′},$ and $DD_{′′}$ are all parallel, congruent, and have the same direction.

In the previous example, it was concluded that the composition of reflections in parallel lines gives the same result as a translation. This conclusion is not an isolated fact. Actually, there is a theorem that guarantees this result.

The composition of two reflections across parallel lines is a single translation. Furthermore, the translation vector is perpendicular to both parallel lines, and its magnitude is twice the distance between the parallel lines.

In the diagram, $△ABC$ is first reflected across $ℓ$ and then the image is reflected across $m.$ Equivalently, the following statements hold true.

- Triangle $A_{′′}B_{′′}C_{′′}$ is a translation of triangle $ABC$ along $v.$
- Vector $v$ is perpendicular to $ℓ$ and $m.$
- The magnitude of $v$ is $2d,$ where $d$ is the distance between $ℓ$ and $m.$

The proof will be developed focusing the attention on vertex $C$ and its images, but the conclusions are true for all the vertices. First, start reflecting $△ABC$ across $ℓ.$ By definition of reflection, the line $ℓ$ is the perpendicular bisector of $CC_{′}.$ Let $P_{1}$ be the intersection point of $ℓ$ and $CC_{′}.$

Next, perform the reflection of $△A_{′}B_{′}C_{′}$ across line $m.$ This time, the line $m$ is the perpendicular bisector of $C_{′}C_{′′}.$ Let $P_{2}$ be the intersection point of $m$ and $C_{′}C_{′′}.$

Since $ℓ$ and $m$ are parallel lines and $CC_{′}$ is perpendicular to $ℓ,$ by the Perpendicular Transversal Theorem, $CC_{′}$ is perpendicular to $m.$ Also, $C_{′}C_{′′}$ is perpendicular to $m.$ Consequently, $CC_{′}$ and $C_{′}C_{′′}$ are parallel vectors with a common point. Therefore, these vectors belong to the same line.

The points $C,$ $C_{′},$ and $C_{′′}$ are collinear.

$CC_{′′} =CC_{′}+C_{′}C_{′′}=(CP_{1}+P_{1}C_{′})+(C_{′}P_{2}+P_{2}C_{′′}) $

Remember that, by definition of reflection, $CP_{1}=P_{1}C_{′}$ and $C_{′}P_{2}=P_{2}C_{′′}.$ Substitute these values into the previous equation.
$CC_{′′}=CP_{1}+P_{1}C_{′}+C_{′}P_{2}+P_{2}C_{′′}$

Substitute values and evaluate

SubstituteII

$CP_{1}=P_{1}C_{′}$, $C_{′}P_{2}=P_{2}C_{′′}$

$CC_{′′}=P_{1}C_{′}+P_{1}C_{′}+C_{′}P_{2}+C_{′}P_{2}$

AddTerms

Add terms

$CC_{′′}=2P_{1}C_{′}+2C_{′}P_{2}$

FactorOut

Factor out $2$

$CC_{′′}=2(P_{1}C_{′}+C_{′}P_{2})$

Segment Addition Postulate

$CC_{′′}=2P_{1}P_{2}$

$AA_{′′}⊥ℓBB_{′′}⊥ℓCC_{′′}⊥ℓ andandand AA_{′′}⊥mBB_{′′}⊥mCC_{′′}⊥m $

From these relations, it is obtained that $AA_{′′},$ $BB_{′′},$ and $CC_{′′}$ are parallel segments. Additionally, all these segments have the same length.
$AA_{′′}BB_{′′}CC_{′′} =2d=2d=2d $

Last but not least, notice that $AA_{′′},$ $BB_{′′},$ and $CC_{′′}$ all have the same direction, which is the same as pointing from the first line to the second line. Consequently, $△A_{′′}B_{′′}C_{′′}$ is a translation of $△ABC$ along $v,$ where $v$ is either $AA_{′′},$ $BB_{′′},$ or $CC_{′′}.$
Notice that the vertices of the preimage can be positioned differently relatively to the lines of reflection. Here, the proof was developed for $C$ positioned not too far to the left of $ℓ,$ which resulted in $C_{′}$ lying between the two lines. However, there are several other possibilities.

The same is true for vertices $A$ and $B.$ Nevertheless, the claim is true for any case. One interesting fact here is that the translation vector always points in the same direction as pointing from the first line to the second line.

In the diagram above, $△ABC$ was first reflected across $m$ and then reflected across $ℓ.$

Additionally, there is also a theorem for the case where the lines of reflection intersect each other.

The composition of two reflections across intersecting lines is a single rotation. Additionally, the center of rotation is the point of intersection of the lines, and the angle of rotation is twice the measure of the acute or right angle formed by the lines.

In the diagram, $△ABC$ is first reflected across $ℓ$ and then the image is reflected across $m.$ The same result is obtained when $△ABC$ is rotated by an angle of $2α_{∘}$ about point $P.$

Let $P$ be the intersection point between the lines $ℓ$ and $m.$ To prove that $△A_{′′}B_{′′}C_{′′}$ is a rotation of $△ABC,$ the following two facts will be proved.

- The point $P$ is the same distance from a vertex of $△ABC$ as it is from the corresponding vertex of $△A_{′′}B_{′′}C_{′′}.$ That is, $PA=PA_{′′},$ $PB=PB_{′′},$ and $PC=PC_{′′}.$
- The angles $∠APA_{′′},$ $∠BPB_{′′},$ and $∠CPC_{′′}$ have all a measure of $2α,$ where $α$ is the acute angle formed by the lines of reflection.

By definition of reflection, the line $ℓ$ is the perpendicular bisector of the segments $AA_{′},$ $BB_{′},$ and $CC_{′}.$

Since $P$ is on $ℓ,$ the Perpendicular Bisector Theorem guarantees that $P$ is equidistant from the endpoints of $AA_{′},$ $BB_{′},$ and $CC_{′}.$$PAPBPC =PA_{′}=PB_{′}=PC_{′} $

Similarly, the line $m$ is the perpendicular bisector of $A_{′}A_{′′},$ $B_{′}B_{′′},$ and $C_{′}C_{′′}.$ Therefore, $P$ is equidistant from the endpoints of these segments.
In consequence, $PA_{′}=PA_{′′},$ $PB_{′}=PB_{′′},$ and $PC_{′}=PC_{′′}.$ Finally, the Transitive Property of Equality can be used to obtain the first part of the proof.

That way, it has been shown that $P$ is the same distance from a vertex of $△ABC$ as it is from the corresponding vertex of $△A_{′′}B_{′′}C_{′′}.$

Here, it will be shown that $m∠APA_{′′}$ is $2α,$ where $α$ is the acute angle formed by the lines. Let $Q$ and $T$ be the intersection points between $AA_{′}$ and $ℓ,$ and $A_{′}A_{′′}$ and $m$ respectively.

Next, consider the right triangles $PQA$ and $PQA_{′}.$ Notice that their hypotenuses $PA$ and $PA_{′}$ are congruent as well as their legs $AQ $ and $A_{′}Q .$

Therefore, $△PQA$ and $△PQA_{′}$ are congruent thanks to the Hypotenuse Leg Theorem. This congruence implies that $∠APQ$ and $∠QPA_{′}$ have the same measure.

$m∠APQ=m∠QPA_{′}(I)$

Similarly, by the Hypotenuse Leg Theorem, $△PTA_{′}$ and $△PTA_{′′}$ are congruent. Consequently, $∠A_{′}PT$ and $∠TPA_{′′}$ are congruent.

$m∠A_{′}PT=m∠TPA_{′′}(II)$

$m∠APA_{′′}=m∠APA_{′}+m∠A_{′}PA_{′′} $

Once more, thanks to the Angle Addition Postulate, each of the two angles on the right-hand side can be rewritten in terms of the angle measures involved in Equations (I) and (II).
$m∠APA_{′′}=m∠APA_{′}+m∠A_{′}PA_{′′}$

SubstituteII

$m∠APA_{′}=$