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## High school geometry

### Course: High school geometry>Unit 1

Lesson 3: 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.
A coordinate plane with point A at four, negative seven. The x- and y- axes scale by one.

### 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.
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, 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.
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 start overline, C, D, end overline will simply be the line segment whose endpoints are C, prime and D, prime.
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, prime, F, prime, G, prime, H, prime, under the translation T, start subscript, left parenthesis, minus, 6, comma, minus, 10, right parenthesis, end subscript.
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.

#### 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.