Virginia SOL 8 - 2020 Edition
5.10 Review: Translations and reflections
Lesson

## Translations

A translation is what occurs when we move an object or shape from one place to another without changing its size, shape or orientation. Sometimes called a slide, a translation moves every point on an object or shape the same distance in the same direction.

To identify and describe a translation

1. Look for corresponding points on the object
2. Identify if the object has moved horizontally or vertically, and then describe that movement as left/right or up/down
3. Count the number of places (or units) that the object has moved

#### Worked example

##### Question 1

Describe the translation from the preimage A to the image B.

Think: First, we need to look for corresponding points on the preimage and image. Next we need to identify the direction the preimage was moved. Finally, we need to state how far it moved in each direction.

Do:

• Notice the highlighted pair of corresponding points in yellow
• We can see that the shape has a horizontal translation and that the preimage A has to move left to become image B.
• Counting the units using the corresponding points we can see that object A has moved 3 units.

So the translation from A to B is 3 units left.

Reflect: If we know how we got from A to B, can we quickly determine how to go from B back to A?

#### Exploration

This applet shows the translation of an object to its image.  You can use the sliders to change the horizontal and vertical amounts.

Summary

If we translate a point or object up or down, then the $x$x -values of the coordinates remain the same and the $y$y -values increase or decrease by the number of units it was translated.

If we translate a point or object right or left, then the $y$y -values of the coordinates remain the same and the $x$x -values increase or decrease by the number of units it was translated.

#### Practice questions

##### Question 2

What is the translation from triangle $B$B to triangle $A$A?

1. $2$2 units left

A

$2$2 units right

B

$3$3 units left

C

$3$3 units right

D

$2$2 units left

A

$2$2 units right

B

$3$3 units left

C

$3$3 units right

D

##### Question 3

What is the translation from square $B$B to square $A$A?

1. $3$3 units up

A

$3$3 units down

B

$4$4 units up

C

$4$4 units down

D

$3$3 units up

A

$3$3 units down

B

$4$4 units up

C

$4$4 units down

D

##### Question 4

1. $2$2 units left and $4$4 units down

A

$4$4 units left and $2$2 units down

B

$2$2 units right and $4$4 units up

C

$4$4 units right and $2$2 units up

D

$2$2 units left and $4$4 units down

A

$4$4 units left and $2$2 units down

B

$2$2 units right and $4$4 units up

C

$4$4 units right and $2$2 units up

D

### Reflections

A reflection is what occurs when we flip an object or shape across a line.  Like a mirror, the object is exactly the same size, just flipped in position.  So what was on the left may now appear on the right.  Every point on the object or shape has a corresponding point on the image, and they will both be the same distance from the reflection line.

Have a quick play with this interactive. Here you can change the shape of the object and the position of the mirror line.

#### Practice questions

##### Question 5

Which image below represents the reflection for the following shape about the vertical mirror line?

1. A

B

C

D

A

B

C

D

##### Question 6

The square below is reflected about the given line. Which point would correspond to point $A$A?

1. H.

A

G.

B

I.

C

F.

D

H.

A

G.

B

I.

C

F.

D

### Reflections across an axis

As we saw above, a reflection occurs when we flip an object or shape across a line like a mirror. We can reflect points, lines, or polygons on a graph by flipping them across an axis or another line in the plane.

Reflecting over the $y$y-axis

Note how the point $\left(-2,1\right)$(2,1) becomes $\left(2,1\right)$(2,1).  The $y$y -value has stayed the same while the $x$x -value has changed signs.In this diagram, the image is reflected across $y$y -axis.

Similarly the point $\left(-6,3\right)$(6,3) becomes $\left(6,3\right)$(6,3).  The $y$y -value have stayed the same and the $x$x -value has changed signs.

Reflecting over the $x$x-axis

Note how the point $\left(4,3\right)$(4,3) becomes $\left(4,-3\right)$(4,3).  The $x$x -value has stayed the same and the $y$y -value has changed signs.

Similarly, the point $\left(0,5\right)$(0,5) becomes $\left(0,-5\right)$(0,5).  The $x$x -value have stayed the same and the $y$y -values has changed signs.

Summary

If we reflect horizontally across the $y$y -axis, then the $y$y -values of the coordinates remain the same and the $x$x -values change sign.

If we reflect vertically across the $x$x -axis, the $x$x -values of the coordinates will remain the same and the $y$y -values will change sign.

#### Practice questions

##### Question 7

Plot the following.

1. Plot the point $A$A$\left(2,-2\right)$(2,2).

2. Now plot point $A'$A, which is a reflection of point $A$A about the $x$x-axis.

##### Question 8

Plot the following.

1. Plot the line segment $AB$AB, where the endpoints are $A$A$\left(-6,-1\right)$(6,1) and $B$B$\left(10,8\right)$(10,8).

2. Now plot the reflection of the line segment about the $y$y-axis.

## Multiple transformations - reflections and translations

A composition of transformations is a list of transformations that are performed one after the other. For example, we might first translate a shape in some direction, then reflect that shape across a line. The first transformation is the translation, the second transformation is the reflection, and the composition is the combination of the two.

#### Exploration

The rectangle below has vertices labeled $ABCD$ABCD. Let's perform a composition of transformations involving a translation followed by a reflection.

Rectangle $ABCD$ABCD starts in the 3rd quadrant.

First, let's translate the rectangle $5$5 units to the left and $11$11 units up. This translated rectangle will have vertices labeled $A'B'C'D'$ABCD.

Next we'll reflect the rectangle $A'B'C'D'$ABCD across the $y$y-axis to produce the rectangle $A"B"C"D"$A"B"C"D". Both transformations are shown on the number plane below.

Translating $ABCD$ABCD to get $A'B'C'D'$ABCD, then reflecting to get $A"B"C"D"$A"B"C"D".

The number of dashes on each vertex of the shape allows us to keep track of the number and order of transformations. Notice that if we reverse the order of the composition we get a different result after both transformations.

The order of transformations is important.

This is the case for compositions in general, although there are some special compositions for which the order does not matter.

#### Practice questions

##### Question 9

The given triangle is to undergo two transformations.

1. First, plot the triangle that results from reflecting the given triangle across the $x$x-axis.

2. The original triangle and the reflected triangle from the previous part are given. Now plot the triangle that results when the original triangle is reflected across the $x$x-axis and translated $4$4 units right.

##### Question 10

Plot the triangle that will result when the original triangle is reflected across the $x$x-axis and translated $3$3 units to the right. Ensure that you have performed both transformations before submitting your answer.

##### Question 11

Points $A\left(-5,-7\right)$A(5,7), $B\left(4,4\right)$B(4,4) and $C\left(9,1\right)$C(9,1) are the vertices of a triangle. What are the coordinates $A'$A, $B'$B and $C'$C that result from reflecting the triangle across the $y$y-axis, and translating it $3$3 units right and $5$5 units up?

1. Ensure that you have performed both transformations before submitting your answer.

$A'\left(\editable{},\editable{}\right)$A(,)

$B'\left(\editable{},\editable{}\right)$B(,)

$C'\left(\editable{},\editable{}\right)$C(,)

### Outcomes

#### 8.7a

Given a polygon, apply transformations, to include translations, reflections, and dilations, in the coordinate plane