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### Course: Integrated math 1>Unit 15

Lesson 2: Translations

# Translating shapes

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 ${T}_{\left(a,b\right)}$.

## Part 1: Translating points

### Let's study an example problem

Find the image ${A}^{\prime }$ of $A\left(4,-7\right)$ under the transformation ${T}_{\left(-10,5\right)}$.

### Solution

The translation ${T}_{\left(-10,5\right)}$ 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 $A\left(4,-7\right)$.
We can also find ${A}^{\prime }$ algebraically:
${A}^{\prime }=\left(4-10,-7+5\right)=\left(-6,-2\right)$

#### Problem 1

Draw the image of $B\left(6,2\right)$ under transformation ${T}_{\left(-4,-8\right)}$.

#### Problem 2

What is the image of $\left(23,-15\right)$ under the translation ${T}_{\left(12,32\right)}$?
$\left($
$,$
$\right)$

## Part 2: Translating line segments

### Let's study an example problem

Consider line segment $\stackrel{―}{CD}$ drawn below. Let's draw its image under the translation ${T}_{\left(9,-5\right)}$.

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

## Part 3: Translating polygons

### Let's study an example problem

Consider quadrilateral $EFGH$ drawn below. Let's draw its image, ${E}^{\prime }{F}^{\prime }{G}^{\prime }{H}^{\prime }$, under the translation ${T}_{\left(-6,-10\right)}$.

### Solution

When we translate a polygon, we are actually translating all the individual line segments that make up that polygon!
Basically, what we did here is to find the images of $E$, $F$, $G$, and $H$ and connect those image vertices.

#### Problem 1

Draw the image of $\mathrm{△}IJK$ under the translation ${T}_{\left(-5,2\right)}$.

#### Problem 2

Draw the images of $\stackrel{―}{LM}$ and $\stackrel{―}{NO}$ under the translation ${T}_{\left(10,0\right)}$.

#### Challenge problem

The translation ${T}_{\left(4,-7\right)}$ mapped $\mathrm{△}PQR$ to an image. The image, $\mathrm{△}{P}^{\prime }{Q}^{\prime }{R}^{\prime }$, is drawn below.
Draw $\mathrm{△}PQR$.

## Want to join the conversation?

• I was confused on the last one. Like I moved it 4,-7 but it's still wrong. Can someone help me please?
• The issue is that the question asks you to go from the image back to the preimage. So if the translation from the preimage to the image is <4,-7>, then to go backwards from the image to the preimage, you have to go backwards along the vector <-4,7>.
• How do you construct a translation with a compass with a point away from the shape?
• I need to understand problem 2 on translating line LM and NO
• Here is a strategy:

Move the green line onto the blue line so that you could see it overlapping it. Then move one point of the green line depending on the translation. In problem 2, the translation is (10,0), so it means that you have to move the point 10 units to the right, and 0 units up. Then do the same thing on the other point. Then you are done.
• I have no idea how to do a reverse translation...
• flip the signs, so -4 turns into +4 and 7 into -7
• I was confused on the direction
• Does that mean we are mapping the original triangle in this question?
• Yes. As one can see, the question shows P'Q'R', and asks you to map PQR. Remember that P'Q'R' is read, "P prime Q prime R prime." "Prime" signifies that it is the IMAGE, or result, of a transformation. So, it wants you to map the original triangle.
• This H.S. Geometry translation helped w/ my spiral translation. My question is: "All of me united is 11 x 6. But when I'm squared my table feet are shown. What am I ?

Ans. Chain squared
• Mind explaining, I don't really understand sorry!
• i dont know any of the questions i need help
• Did you read the instructions?
• if this is never going to be used in our lives why is it taught? to waste time?
• If you become any type of constructor or designer. this will help you figure out how to change the size of what they want or help them figerouthow it can be better.
• I was upset by doing this but I eventually did it.