Introduction to Translations
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Learning text on the topic Introduction to Translations
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 or moving a piece on a chess board. 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 distorting it.
Properties of Translations
- Direction: Translations move points in a straight line.This can be either horizontally, vertically or a combination of both resulting in a diagonal movement.
- 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.
- Vectors: The information given to describe a translation is often given in the form of a vector. This keyword will be explored in different texts, in this text we will focus on the core concepts of translations.
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.
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, rotations 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 it.
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
Introduction to Translations exercise
-
Understand the definition of a translation.
HintsLook 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?
SolutionA 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.
HintsWhen 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'.
SolutionThe new figure is labelled here with E'F'G'H'.
-
Identify the translation a figure underwent.
HintsBe 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'.
SolutionTriangle 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.
HintsOne 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.
HintsA 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.
SolutionThese 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.
HintsDon'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.
SolutionThe translated points will be:
X' = (-1,-2)
Y' = (-3,0)
Z' = (-1,1)
