Learn how to draw the image of a given shape under a given translation.

Introduktion

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\langle a,b \rangle is a transformation that moves all points aa units in the xx-direction and bb units in the yy-direction. Such a transformation is commonly represented as T(a,b)T_{(a,b)}.

Part 1: Translating points

Let's study an example problem

Find the image AA' of A(4,7)A(4,-7) under the transformation T(10,5)T_{(-10,5)}.

Løsning

The translation T(10,5)T_{(\tealD{-10},\maroonD{5})} moves all points 10\tealD{-10} in the xx-direction and +5\maroonD{+5} in the yy-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)A(4,-7).
We can also find AA' algebraically:
A=(410,7+5)=(6,2)A'=(4\tealD{-10},-7\maroonD{+5})=(-6,-2)

Your turn!

Problem 1

Problem 2

Part 2: Translating line segments

Let's study an example problem

Consider line segment CD\overline{CD} drawn below. Let's draw its image under the translation T(9,5)T_{(9,-5)}.

Løsning

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\overline{CD} will simply be the line segment whose endpoints are CC' and DD'.

Part 3: Translating polygons

Let's study an example problem

Consider quadrilateral EFGHEFGH drawn below. Let's draw its image, EFGHE'F'G'H', under the translation T(6,10)T_{(-6,-10)}.

Løsning

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 EE, FF, GG, and HH and connect those image vertices.

Your turn!

Problem 1

Problem 2

Challenge problem

The translation T(4,7)T_{(4,-7)} mapped PQR\triangle PQR. The image, PQR\triangle P'Q'R', is drawn below.