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## High school geometry

# Determining translations

CCSS.Math:

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.**

### Your turn!

#### Problem 1

#### Problem 2

#### Problem 3

#### 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.

## 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.

#### 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.**

### Your turn!

## 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!(34 votes)- 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(11 votes)

- 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.(12 votes)
- 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!(1 vote)

- 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?(3 votes)
- 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).(3 votes)

- 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(11 votes)- your correct you didn't make a mistake(0 votes)

- Gahhhhh! life is too hard!(8 votes)
- I understand. Life can get tough but if you think really hard and believe, you can do it.(3 votes)

- I dont get how to do this it is very confusing to me(8 votes)
- Where are you getting confused? You are trying to find the vertex from a preimage point (no tic mark) to an image point (with a prime), so you subtract the <x'-x,y'-y> to get a vector, and once you find the vector, all points of a figure will be translated along the same vector. Take the last example using J. J is at (2,-4) and J' is at (-2,3). The translation it moved along would be (-2-2,3-(-4)) or (-4,7). Always make sure you know the preimage and the image that you are translating to.(2 votes)

- Can you help me with the translations?(8 votes)
- how come the plot point is different(6 votes)
- If J' moves DOWN seven and to the RIGHT 4 why is the answer -4,7?(4 votes)
- K is the preimage, and K' is the image, so the translation is always from K to K' (pre-image to image) where you appear to want to do it backwards.(0 votes)

- Is the answer to the part 2 question not (7, -4)? I think that's the correct answer but the site is saying it's (8, -4)(2 votes)
- I put in (-4,7) to get it correct. From any point, you go -4 in the x direction and 7 in the y direction, so neither are correct, but I do not see where you get (8,-4) as the site's answer, I just saw a hint that implies my answer is correct.(2 votes)