Learn how to find the necessary translation to map a given source shape onto a given image shape.
In this article, we will solve problems where we are given starting and ending coordinates and asked to figure out what translation must have occurred.

Part 1: Determining the translation for a single pair of points

Let's study an example problem

A translation maps point A(3,7)A(3,7) to point A(6,2)A'(6,-2). Let's determine what translation this is.

Løsning

Step 1: Horizontal shift. AA is shifted 33 units to the right because (6)(3)=+3(6)-(3)=\tealD{+3}.
Step 2: Vertical shift. AA is shifted 99 units down because (2)(7)=9(-2)-(7)=\maroonD{-9}.
The answer: AA is mapped onto AA' under a translation by 3,9\langle \tealD{3},\maroonD{-9} \rangle.

Your turn!

Problem 1

Problem 2

Problem 3

Challenge problem

A certain translation takes point D(3,10)D(-3,10) to point D(12,21)D'(-12,21).

Part 2: Determining the translation for a pair of polygons

Let's study an example problem

Consider the quadrilaterals drawn below. Let's determine the translation that maps the source FGHI\blueD{FGHI} onto the image FGHI\maroonD{F'G'H'I'}.

Løsning

Let's focus in on a pair of corresponding points, such as F(4,6)F(-4,6) and F(2,3)F'(2,3). If we can find the translation that takes FF to FF', we will necessarily know the translation that takes the entire source quadrilateral to its image!
Horizontal shift: (2)(4)=+6(2)-(-4)=\tealD{+6}
Vertical shift: (3)(6)=3(3)-(6)=\maroonD{-3}
Therefore, FGHIFGHI is mapped onto FGHIF'G'H'I' under a translation by 6,3\langle\tealD{6},\maroonD{-3}\rangle.

Your turn!