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## Basic geometry and measurement

### Course: Basic geometry and measurement>Unit 14

Lesson 2: Translations

# Determining translations

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, left parenthesis, 3, comma, 7, right parenthesis to point A, prime, left parenthesis, 6, comma, minus, 2, right parenthesis. Let's determine what translation this is.

#### Solution

Step 1: Horizontal shift. A is shifted 3 units to the right because left parenthesis, 6, right parenthesis, minus, left parenthesis, 3, right parenthesis, equals, start color #01a995, plus, 3, end color #01a995.
Step 2: Vertical shift. A is shifted 9 units down because left parenthesis, minus, 2, right parenthesis, minus, left parenthesis, 7, right parenthesis, equals, start color #ca337c, minus, 9, end color #ca337c.
The answer: A is mapped onto A, prime under a translation by open angle, start color #01a995, 3, end color #01a995, comma, start color #ca337c, minus, 9, end color #ca337c, close angle.

#### Problem 1

Determine the translation that maps point B, left parenthesis, 2, comma, 1, right parenthesis to point B, prime, left parenthesis, minus, 4, comma, 5, right parenthesis.
open angle
comma
close angle

#### Problem 2

Determine the translation that maps point C, left parenthesis, 7, comma, 5, right parenthesis to point C, prime, left parenthesis, 5, comma, 5, right parenthesis.
open angle
comma
close angle

#### Problem 3

In general, which calculation gives the exact vertical shift of a translation from point P to point P, prime?

#### Challenge problem

A certain translation takes point D, left parenthesis, minus, 3, comma, 10, right parenthesis to point D, prime, left parenthesis, minus, 12, comma, 21, right parenthesis.
What is the image of E, left parenthesis, 17, comma, minus, 9, right parenthesis under this translation?
left parenthesis
comma
right parenthesis

## 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 pre-image start color #11accd, F, G, H, I, end color #11accd onto the image start color #ca337c, F, prime, G, prime, H, prime, I, prime, end color #ca337c.
A coordinate plane. The x- and y-axes both scale by one. Trapezoid FGHI with point F at negative four, six, point G at one, six, point H at zero, two, and point I at negative four, two. Trapezoid F prime, G prime, H prime, I prime with points F prime at two, three, G prime at seven, three, H prime at six, negative one, and I prime two, negative one.

#### Solution

Let's focus in on a pair of corresponding points, such as F, left parenthesis, minus, 4, comma, 6, right parenthesis and F, prime, left parenthesis, 2, comma, 3, right parenthesis. If we can find the translation that takes F to F, prime, we will necessarily know the translation that takes the entire pre-image quadrilateral to its image!
Horizontal shift: left parenthesis, 2, right parenthesis, minus, left parenthesis, minus, 4, right parenthesis, equals, start color #01a995, plus, 6, end color #01a995
Vertical shift: left parenthesis, 3, right parenthesis, minus, left parenthesis, 6, right parenthesis, equals, start color #ca337c, minus, 3, end color #ca337c
Therefore, F, G, H, I is mapped onto F, prime, G, prime, H, prime, I, prime under a translation by open angle, start color #01a995, 6, end color #01a995, comma, start color #ca337c, minus, 3, end color #ca337c, close angle.