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## High school geometry

# Translating shapes

CCSS.Math:

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:

### Your turn!

#### Problem 1

#### Problem 2

## 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.

### Your turn!

#### Problem 1

#### Problem 2

#### 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.

## Want to join the conversation?

- This should be less challenging(15 votes)
- I mean unless you know another easier way there is nothing you can do except learnt it(unless you go and ask the universe to change its rules).(6 votes)

- How do you construct a translation with a compass with a point away from the shape?(2 votes)
- why was this made this is to hard curse new math(10 votes)
- why doesnt math slay(5 votes)
- aint no way you just said that(5 votes)

- I need to understand problem 2 on translating line LM and NO(4 votes)
- 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.(2 votes)

- if this is never going to be used in our lives why is it taught? to waste time?(3 votes)
- 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.(3 votes)

- How did I even get here? I thought this was supposed to teach me how to draw polygons with coordinates? GAHH MATH IS SO HARD!(4 votes)
- No this is about moving stuff around. For graphing polygons just plot all points and connect the dots(1 vote)

- this makes no sense(4 votes)
- I dont understand that well without the graph(2 votes)
- Just watch this video https://www.khanacademy.org/math/basic-geo/basic-geo-transformations-congruence/basic-geometry-translations/v/translating-points

it explains how to do it without a graph at0:53.(3 votes)

- The last two questions are a bit confusing? i dont quite understand(2 votes)