# Introduction to Translations

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## Translations in Maths – Definition

In our daily lives, we often shift objects from one place to another without changing their shape or size, like sliding a book across a table. In geometry, this movement is known as a translation. Before reading on, you can have a look at these videos to remind yourselves about coordinates in the first quadrant Positions in the First Quadrant and Plotting Shapes on a Grid.

A translation in geometry is a transformation (or the movement of a shape) that moves every point of a figure the same distance in the same direction. It slides an object from one position to another without rotating, resizing or deforming it.

## Properties of Translations

• Direction: Translations move points in a straight line.
• Distance: Each point moves the same distance.
• Congruency: The original shape and its translation are congruent. This means they are exactly the same shape and size.

To have the most success with this topic, it is important that you are familiar with Coordinates in All Four Quadrants.

## Translations – Step-by-Step Process

Below we have a triangle with vertices A(1,2), B(3,2), and C(1,4), and we want to translate it 2 units to the right and 3 units down.

Step Description Example
1 Identify Original Coordinates A$(1,2)$, B$(3,2)$, C$(1,4)$
2 Determine the Translation 2 units right, 3 units down
3 Apply the Translation to Each Point A': $(1+2, 2-3) = (3, -1)$
B': $(3+2, 2-3) = (5, -1)$
C': $(1+2, 4-3) = (3, 1)$
4 Plot and Label the Translated Figure
5 Verify Congruency Check that original and translated figures are the same shape and size

## Translation Notation

In translation notation, we describe exactly how to shift each point. We use $(x, y) \rightarrow (x+a, y+b)$, where:

• $x, y$ are the original points.
• $a$ is how far we move left or right.
• $b$ is how far we move up or down.

For example, $(x, y) \rightarrow (x+3, y-2)$ means move every point 3 units right and 2 units down.

Original Point Translation Notation Translated Point
$(2, 3)$ $\rightarrow (x+4, y+1)$ $(6, 4)$
$(5, 5)$ $\rightarrow (x-3, y-2)$ $(2, 3)$
$(1, 2)$ $\rightarrow (x+2, y+5)$ $(3, 7)$
$(4, 4)$ $\rightarrow (x+1, y-1)$ $(5, 3)$
$(3, 1)$ $\rightarrow (x-2, y+3)$ $(1, 4)$

### Labelling a Translated Figure

When we move geometric shapes, like sliding a square on graph paper, we use special labels to keep track of the new positions.

• Easy to Spot: By adding a little mark, called a 'prime' (like A'), we can see which points have moved. It's like tagging something with a sticky note.
• Matching Points: Labels help us match the original points with their new spots. If point A moves, it becomes A'. This way, we know A and A' are related.
• Simple and Clear: Using labels is an easy way to show which shape has moved and where it has gone. It keeps our diagrams tidy and easy to understand.

Using labels is like putting a "Moved!" sign on our shapes. It's a simple, clear way to show where they've gone.

## Translations – Guided Practice

Let’s work through the following translation of a triangle on a coordinate grid.

Identify the coordinates of the triangle's vertices. Label them X, Y and Z.
Translate the triangle 3 units right and 2 units down.
Verify if the original and translated triangles are congruent and label the translated triangle.

Identify the coordinates of the rectangle's vertices. Label them A, B, C and D.
Translate the rectangle following the rule: $T\left(x+2, y+1\right)$.
What are the new coordinates after translation?
Verify if the original and translated rectangles are congruent.

Not only can you perform isolated translations, but also a sequence of translations can be done to move a figure further, while still keeping it congruent.

## Translations – Exercises

To practise translating, find a sheet of graph paper to practise with the following exercises.

For some more advanced transformations, sequences of reflections and translations can be done to a figure, and it will still be congruent!

## Translations – Summary

Key Learnings from this Text:

• Translations slide a shape in a straight line without rotating or resizing.

• Each point in the shape moves the same distance and direction.

• The original and translated shapes are congruent or exactly the same shape and size.

• Understanding translations helps visualise how shapes interact within a space.

For more, have a look at Translations on a Grid.

## Translations – Frequently Asked Questions

What is a translation in geometry?
Are translated shapes always the same size?
How do you describe a translation?
Can translations change the orientation of a shape?
Why are translations important in geometry?
What is an example of a translation in real life?
How do you perform a translation on a coordinate grid?
What does it mean for two shapes to be congruent?
Can translations be reversed?
How does translation differ from rotation or reflection?

## Introduction to Translations exercise

Would you like to apply the knowledge you’ve learnt? You can review and practice it with the tasks for the learning text Introduction to Translations.
• ### Understand the definition of a translation.

Hints

Look at the words you know in the definition to help you find the meaning of a translation.

• resize
• flip
• turn
• slide

Look carefully at figure ABC in the header. What do you notice about figure A'B'C'?

• It is a different size?
• Did the image flip?
• Has it been turned?
• Did the figure slide?
Solution

A translation is sliding a shape across a plane without rotating or flipping it.

When shapes are translated, they can go left or right, and up or down. In the example seen here, the image was translated 9 units down.

• ### Identify labels and coordinates of a translated figure.

Hints

When we move geometric shapes, like sliding a square on graph paper, we use special labels to keep track of the new positions like in this example.

In this example you can see that figure ABC was translated and is now labelled A'B'C'.

Solution

The new figure is labelled here with E'F'G'H'.

• ### Identify the translation a figure underwent.

Hints

Be sure to count each unit carefully when determining the translation. Each coordinate moved on the graph is considered 1 unit.

By adding a little mark, (like A'), we can see which points have moved.

Remember to find the rule of each point translated to its corresponding point. For example, point A is translated to A'.

Solution

Triangle ABC was translated Right 5 units, Down 3 units to triangle A'B'C'.

• ### Apply your knowledge of translations to translate a point based on the directions.

Hints

One way to translate a coordinate is to count in the given direction on a graph.

$(-2,3)$ translated 4 units right and 2 units down ends up on $(2,1)$.

Another way to translate a coordinate is to add or subtract the $x$ and $y$ values.

$(-2,3)$ translated 4 units right and 2 units down $(-2+4, 3-2)\rightarrow(2,1)$

Solution
• Original: (-4, 2) Translated 4 units right, 2 units down: (0, 0)
• Original: (-4, 0) Translated 4 units right, 2 units down: (0, -2)
• Original: (-1, 0) Translated 4 units right, 2 units down: (3, -2)
• Original: (-1, 2) Translated 4 units right, 2 units down: (3, 0)
• ### Identify which transformation is a translation.

Hints

A translation is when you slide a shape to a different spot without turning or flipping it.

Imagine moving a piece of paper across your desk without turning it over or spinning it around. The shape still looks the same, just in a new place.

In the image here figure ABCD has been translated right 6 units and up 2 units to A'B'C'D'.

When a shape is translated, the size and orientation of the shape stay the same.

Solution

These triangles have been translated because they can slide from one to the other without changing directions or size. These shapes are congruent.

• ### Translate a figure and label the new coordinates.

Hints

Don't forget to add a comma (,) between the coordinates in the translated figure.

It may help you to draw this on graph paper and count the units to find the translated figure.

You can also use the translation notation to calculate the coordinates of the translated figure.

The rule, $\rightarrow (x-1, y-2)$, means that the figure will move left 1 unit and then down 2 units.

Solution

The translated points will be:

X' = (-1,-2)

Y' = (-3,0)

Z' = (-1,1)

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