Translating shapes (article) | Translations

Learn how to draw the image of a given shape under a given translation.

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 $⟨ a, b ⟩$ ‍ 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 $T_{(- 10, 5)}$ ‍.

A coordinate plane with point A at four, negative seven. The x- and y- axes scale by one.

Solution

The translation $T_{(- 10, 5)}$ ‍ moves all points $- 10$ ‍ in the $x$ ‍-direction and $+ 5$ ‍ 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 $A (4, - 7)$ ‍.

A coordinate plane with point A at four, negative seven. The x- and y- axes scale by one. A dashed arrow points ten units to the left and up five units to point A prime at negative six, negative two.

We can also find $A^{'}$ ‍ algebraically:

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

Your turn!

Problem 1

Draw the image of $B (6, 2)$ ‍ under transformation $T_{(- 4, - 8)}$ ‍.

The translation $T_{(- 4, - 8)}$ ‍ moves all points $- 4$ ‍ in the $x$ ‍-direction and $- 8$ ‍ in the $y$ ‍-direction. In other words, it moves everything 4 units to the left and 8 units down.

Problem 2

What is the image of $(23, - 15)$ ‍ under the translation $T_{(12, 32)}$ ‍?

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The translation $T_{(12, 32)}$ ‍ moves all points $+ 12$ ‍ in the $x$ ‍-direction and $+ 32$ ‍ in the $y$ ‍-direction.

Part 2: Translating line segments

Let's study an example problem

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

A coordinate plane with a line segment with endpoints C at negative seven, eight to D at negative four, one. The x- and y- axes scale by one.

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.

A coordinate plane with a line segment with endpoints C at negative seven, eight to D at negative four, one. The x- and y- axes scale by one. An arrow points nine units to the right from C and down five units to point C prime. An arrow points nine units to the right from D and down five units to point D prime.

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

A coordinate plane with a line segment with endpoints C at negative seven, eight to D at negative four, one. The x- and y- axes scale by one. Another line segment has endpoints C prime at two, three and D prime at five, negative four. An arrow points from endpoint C to endpoint C prime and another arrow points from endpoint D to D prime.

Part 3: Translating polygons

Let's study an example problem

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

A coordinate plane with a quadrilateral with vertices E at negative one, six, F at three, eight, G at two, two, and H at negative two, three. The x- and y- axes scale by one.

Solution

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

A coordinate plane with a quadrilateral with vertices E at negative one, six, F at three, eight, G at two, two, and H at negative two, three. The x- and y- axes scale by one. A congruent quadrilateral with has vertices E prime at negative seven, negative four, F prime at negative three, negative two, G prime at negative four, negative eight, and H prime at negative eight, negative seven. An arrow points from vertex E to E prime. An arrow points from vertex F to F prime. Another arrow points from G to G prime, and an arrow points from vertex H to H prime.

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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Translating shapes

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Your turn!

Problem 1

Draw the image of $△ I J K$ ‍ under the translation $T_{(- 5, 2)}$ ‍.

The translation $T_{(- 5, 2)}$ ‍ moves all points $- 5$ ‍ in the $x$ ‍-direction and $+ 2$ ‍ in the $y$ ‍-direction. In other words, it moves everything 5 units to the left and 2 units up.

Problem 2

Draw the images of $\overset{―}{L M}$ ‍ and $\overset{―}{N O}$ ‍ under the translation $T_{(10, 0)}$ ‍.

The translation $T_{(10, 0)}$ ‍ moves all points $+ 10$ ‍ in the $x$ ‍-direction and $0$ ‍ in the $y$ ‍-direction. In other words, it moves everything 10 units to the right without any vertical movement.

Challenge problem

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

Draw $△ P Q R$ ‍.

We know that under the translation $T_{(4, - 7)}$ ‍, the original triangle $△ P Q R$ ‍ moved 4 units to the right and 7 units down to obtain $△ P^{'} Q^{'} R^{'}$ ‍.

Therefore, the original triangle should be 4 units to the left and 7 units up from $△ P^{'} Q^{'} R^{'}$ ‍.

Translating shapes (article) | Translations | Khan Academy (2024)

Introduction

Part 1: Translating points

Let's study an example problem

Solution

Your turn!

Problem 1

Problem 2

Part 2: Translating line segments

Let's study an example problem

Solution

Part 3: Translating polygons

Let's study an example problem

Solution

Your turn!

Problem 1

Problem 2

Challenge problem