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Course: Geometry (all content) > Unit 10

Lesson 2: Translations
Math
Geometry (all content)
Transformations
Translations
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Translating shapes

Google Classroom
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).

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).
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).

Problem 2

What is the image of (23,−15) under the translation T(12,32)?
(
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)

Part 2: Translating line segments

Let's study an example problem

Consider line segment CD― drawn below. Let's draw its image under the translation T(9,−5).

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 CD― will simply be the line segment whose endpoints are C′ and D′.

Part 3: Translating polygons

Let's study an example problem

Consider quadrilateral EFGH drawn below. Let's draw its image, E′F′G′H′, under the translation T(−6,−10).

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

Draw the image of △IJK under the translation T(−5,2).

Problem 2

Draw the images of LM― and NO― under the translation T(10,0).

Challenge problem

The translation T(4,−7) mapped △PQR to an image. The image, △P′Q′R′, is drawn below.
Draw △PQR.

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