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## High school geometry

### Course: High school geometry>Unit 1

Lesson 3: 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 $A\left(3,7\right)$ to point ${A}^{\prime }\left(6,-2\right)$. Let's determine what translation this is.

#### Solution

Step 1: Horizontal shift. $A$ is shifted $3$ units to the right because $\left(6\right)-\left(3\right)=+3$.
Step 2: Vertical shift. $A$ is shifted $9$ units down because $\left(-2\right)-\left(7\right)=-9$.
The answer: $A$ is mapped onto ${A}^{\prime }$ under a translation by $⟨3,-9⟩$.

#### Problem 1

Determine the translation that maps point $B\left(2,1\right)$ to point ${B}^{\prime }\left(-4,5\right)$.
$⟨$
$,$
$⟩$

#### Problem 2

Determine the translation that maps point $C\left(7,5\right)$ to point ${C}^{\prime }\left(5,5\right)$.
$⟨$
$,$
$⟩$

#### 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 $D\left(-3,10\right)$ to point ${D}^{\prime }\left(-12,21\right)$.
What is the image of $E\left(17,-9\right)$ under this translation?
$\left($
$,$
$\right)$

## 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 $FGHI$ onto the image ${F}^{\prime }{G}^{\prime }{H}^{\prime }{I}^{\prime }$.

#### Solution

Let's focus in on a pair of corresponding points, such as $F\left(-4,6\right)$ and ${F}^{\prime }\left(2,3\right)$. 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: $\left(2\right)-\left(-4\right)=+6$
Vertical shift: $\left(3\right)-\left(6\right)=-3$
Therefore, $FGHI$ is mapped onto ${F}^{\prime }{G}^{\prime }{H}^{\prime }{I}^{\prime }$ under a translation by $⟨6,-3⟩$.

Determine the translation that maps $\mathrm{△}JKL$ onto $\mathrm{△}{J}^{\prime }{K}^{\prime }{L}^{\prime }$.
$⟨$
$,$
$⟩$

## 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).
• 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!
• 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 ?
• I don't get this
• You translate it think jared think
• 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