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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.

Determine the translation that maps start color #11accd, triangle, J, K, L, end color #11accd onto start color #ca337c, triangle, J, prime, K, prime, L, prime, end color #ca337c.
A coordinate plane with a triangle with vertices J at two, negative four, K at eight, negative three, and L at six, negative eight. The x- and y- axes scale by one. A congruent triangle has vertices J prime at negative two, three, K prime at four, four, and L prime at two, negative one.
open angle
comma
close angle

## Want to join the conversation?

• I don't get this problem:

Challenge problem

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

What is the image of E(17, -9) under this translation?

I tried (-9, 11), since you need -9 to from -3 to -12, and 11 since it's 11 from 10 to 21; I also tried (9, -11). Can someone please explain what i'm doing wrong? Thanks!
• You must get the translation by taking the difference between D'(-12,21) and D(-3,-10).Then you will get (-9,11).You must apply this to E(17,-9).So you will get the answer
• The challenge problem is not making any sense, even after watching the video they give you for help. Is there an easier way to understand this or an easier way for it to be explained?
• You have to take the translation from the first problem and add it to the third coordinate. So if it was point A at (7,12) to point B at (11,4). The translation is (4,-8). Now you apply that to point C at (10,15), 10+4, 15-8. So your final point is at (14,7).
• To get the translation vector i can use any points ... for example a object point - any image point = Translation Vector... Well nothing seems to change i get the same result with every point.. So i dont have to do it for every point right? Can i just use one and thats it ?
• If i choose to type only what i got for a certain point, will it be the same as the other points or will I get it wrong? This confuses me.
• It will be the same as all the points moved the same amount. If that makes sense. I just realized this was 5 years ago, oops!
• Gahhhhh! life is too hard!
• I understand. Life can get tough but if you think really hard and believe, you can do it.
• Choosing J, the initial point is at (2,-4). That point then moves to J' at (-2,3)

-2-2=-2+(-2)=-4
3-(-4)=3+4=7

This would make the translation (-4,7). Please help. This is EXTREMELY frustrating and I can't move ahead until I understand where my mistake is
• your correct you didn't make a mistake