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## Geometry (all content)

### Course: Geometry (all content)>Unit 10

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 open angle, a, comma, b, close angle 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, start subscript, left parenthesis, a, comma, b, right parenthesis, end subscript.

## Part 1: Translating points

### Let's study an example problem

Find the image A, prime of A, left parenthesis, 4, comma, minus, 7, right parenthesis under the transformation T, start subscript, left parenthesis, minus, 10, comma, 5, right parenthesis, end subscript.

### Solution

The translation T, start subscript, left parenthesis, start color #01a995, minus, 10, end color #01a995, comma, start color #ca337c, 5, end color #ca337c, right parenthesis, end subscript moves all points start color #01a995, minus, 10, end color #01a995 in the x-direction and start color #ca337c, plus, 5, end color #ca337c 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 parenthesis, 4, comma, minus, 7, right parenthesis.
We can also find A, prime algebraically:
A, prime, equals, left parenthesis, 4, start color #01a995, minus, 10, end color #01a995, comma, minus, 7, start color #ca337c, plus, 5, end color #ca337c, right parenthesis, equals, left parenthesis, minus, 6, comma, minus, 2, right parenthesis

#### Problem 1

Draw the image of B, left parenthesis, 6, comma, 2, right parenthesis under transformation T, start subscript, left parenthesis, minus, 4, comma, minus, 8, right parenthesis, end subscript.

#### Problem 2

What is the image of left parenthesis, 23, comma, minus, 15, right parenthesis under the translation T, start subscript, left parenthesis, 12, comma, 32, right parenthesis, end subscript?
left parenthesis
comma
right parenthesis

## Part 2: Translating line segments

### Let's study an example problem

Consider line segment start overline, C, D, end overline drawn below. Let's draw its image under the translation T, start subscript, left parenthesis, 9, comma, minus, 5, right parenthesis, end subscript.

### 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 start overline, C, D, end overline 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 E, F, G, H drawn below. Let's draw its image, E, prime, F, prime, G, prime, H, prime, under the translation T, start subscript, left parenthesis, minus, 6, comma, minus, 10, right parenthesis, end subscript.

### 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 triangle, I, J, K under the translation T, start subscript, left parenthesis, minus, 5, comma, 2, right parenthesis, end subscript.

#### Problem 2

Draw the images of start overline, L, M, end overline and start overline, N, O, end overline under the translation T, start subscript, left parenthesis, 10, comma, 0, right parenthesis, end subscript.

#### Challenge problem

The translation T, start subscript, left parenthesis, 4, comma, minus, 7, right parenthesis, end subscript mapped triangle, P, Q, R. The image, triangle, P, prime, Q, prime, R, prime, is drawn below.
Draw triangle, P, Q, R.

## Want to join the conversation?

• I dont understand that well without the graph
• I also don't understand this without a graph
• How do you construct a translation with a compass with a point away from the shape?
• ?I dont get it. Can you put it in simpler words?
(1 vote)
• Geometry more like Geomystery because I have no idea what's going on (Joke, this is easy)
• 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>.
• This should be less challenging
• without challenges you would be bored
• 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.
• why doesnt math slay