Translations

In this article, we will solve problems where we are given starting and ending coordinates and asked to figure out what translation must have occurred.

Part 1: Determining the translation for a single pair of points

Let's study an example problem

A translation maps point $A(3,7)$ to point $A'(6,-2)$. Let's determine what translation this is.

Solution

Step 1: Horizontal shift. $A$ is shifted $3$ units to the right because $(6)-(3)=\tealD{+3}$.

Step 2: Vertical shift. $A$ is shifted $9$ units down because $(-2)-(7)=\maroonD{-9}$.

The answer: $A$ is mapped onto $A'$ under a translation by $\langle \tealD{3},\maroonD{-9} \rangle$.

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Your turn!

Problem 1

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Problem 2

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Problem 3

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Challenge problem

A certain translation takes point $D(-3,10)$ to point $D'(-12,21)$.

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Part 2: Determining the translation for a pair of polygons

Let's study an example problem

Consider the quadrilaterals drawn below. Let's determine the translation that maps the pre-image $\blueD{FGHI}$ onto the image $\maroonD{F'G'H'I'}$.

[[☃ image 1]]

Solution

Let's focus in on a pair of corresponding points, such as $F(-4,6)$ and $F'(2,3)$. If we can find the translation that takes $F$ to $F'$, we will necessarily know the translation that takes the entire pre-image quadrilateral to its image!

Horizontal shift: $(2)-(-4)=\tealD{+6}$

Vertical shift: $(3)-(6)=\maroonD{-3}$

Therefore, $FGHI$ is mapped onto $F'G'H'I'$ under a translation by $\langle\tealD{6},\maroonD{-3}\rangle$.

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Your turn!

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Introduction

In this article, we'll practice the art of translating shapes. Mathematically speaking, we will learn how to draw the image of a given shape under a given translation.

A translation by $\langle a,b \rangle$ is a transformation that moves all points $a$ units in the $x$-direction and $b$ units in the $y$-direction. Such a transformation is commonly represented as $T_{(a,b)}$.

Part 1: Translating points

Let's study an example problem

Find the image $A'$ of $A(4,-7)$ under the transformation $T_{(-10,5)}$.

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Solution

The translation $T_{(\tealD{-10},\maroonD{5})}$ moves all points $\tealD{-10}$ in the $x$-direction and $\maroonD{+5}$ in the $y$-direction. In other words, it moves everything 10 units to the left and 5 units up.

Now we can simply go 10 units to the left and 5 units up from $A(4,-7)$.

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We can also find $A'$ algebraically:

$A'=(4\tealD{-10},-7\maroonD{+5})=(-6,-2)$

Your turn!

Problem 1

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Problem 2

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Part 2: Translating line segments

Let's study an example problem

Consider line segment $\overline{CD}$ drawn below. Let's draw its image under the translation $T_{(9,-5)}$.

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Solution

When we translate a line segment, we are actually translating all the individual points that make up that segment.

Luckily, we don't have to translate all the points, which are infinite! Instead, we can consider the endpoints of the segment.

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Since all points move in exactly the same direction, the image of $\overline{CD}$ will simply be the line segment whose endpoints are $C'$ and $D'$.

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Part 3: Translating polygons

Let's study an example problem

Consider quadrilateral $EFGH$ drawn below. Let's draw its image, $E'F'G'H'$, under the translation $T_{(-6,-10)}$.

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Solution

When we translate a polygon, we are actually translating all the individual line segments that make up that polygon!

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Basically, what we did here is to find the images of $E$, $F$, $G$, and $H$ and connect those image vertices.

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Your turn!

Problem 1

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Problem 2

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Challenge problem

The translation $T_{(4,-7)}$ mapped $\triangle PQR$ to an image. The image, $\triangle P'Q'R'$, is drawn below.

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When you translate something in geometry, you're simply moving it around. You don't distort it in any way. If you translate a segment, it remains a segment, and its length doesn't change. Similarly, if you translate an angle, the measure of the angle doesn't change.

These properties may seem obvious, but they're important to keep in mind later on when we do proofs. To make sure we understand these properties, let's walk through a few examples.

Property 1: Line segments are taken to line segments of the same length.

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As you can see for yourself, the pre-image and the image are both line segments with the same length. This is true for any line segment that goes under any translation.

Property 2: Angles are taken to angles of the same measure.

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As you can see for yourself, the pre-image angle and the image angle have the same measure. This is true for any angle that goes under any translation.

Property 3: Lines are taken to lines, and parallel lines are taken to parallel lines.

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As you can see for yourself, each line is taken to another line, and the image lines remain parallel to each other. This is true for any line—or lines—that go under any translation.

Conclusion

We found that translations have the following three properties:

• line segments are taken to line segments of the same length;

• angles are taken to angles of the same measure; and

• lines are taken to lines and parallel lines are taken to parallel lines.

This makes sense because a translation is simply like taking something and moving it up and down or left and right. You don't change the nature of it, you just change its location.

It's like taking the elevator or going on a moving walkway: you start in one place and end in another, but you are the same as you were before, right?