Translation

Key points

An image of a square grid. The grid has a length of fifteen squares and a width of eight squares. Two identical trapeziums have been drawn on the grid. Trapezium B is eight squares to the left of trapezium A. Written left: A to B equals column vector, negative six, zero. — Image caption,
Translations of a shape can be described using a column vector.

A is a type of .

A translation moves a shape from one location to another.
The new shape is to the original shape.
The size of the shape does not change, and the shape is not or .
To understand translation a good understanding of naming and plotting co-ordinates can be helpful.

Describing translation

A translation is described using horizontal and vertical. By counting squares on a grid or set of we can describe a translation. These displacements can then be represented using a column .

A column vector uses two values and curly brackets:$$ \ (^x _y) $$

The $x$ value describes the displacement, moving to the right or left.

A displacement right is a positive number and a displacement left is a negative number.
If there is no horizontal movement a zero is used.

The $y$ value describes the displacement, moving up or down.

A displacement up is a positive number and a displacement down is a negative number.
If there is no vertical movement, a zero is used.

Using a as a reference point helps track the translation of the shape.

Examples

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Image caption,
Decide on a reference point to help describe the translation between shape A and shape B. Here, the reference point is labelled P.
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From shape A to shape B the reference point P has moved 6 to the right. The translation of A to B is described using the vector shown in the image above.
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Using a different point, such as reference point Q, the translation of A to B is still the same vector, or 6 right. This shows the choice of reference point does not matter.
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From shape C to shape D, the reference point P has moved 4 squares down. The translation of shape C to shape D is described as the vector in the image.
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This translation is a combination of horizontal and vertical displacements. Decide on a reference point to help describe the translation. This is labelled P.
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From shape E to shape F, the reference point P has moved 7 squares to the left. This is a horizontal displacement of -7
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It then moves 3 squares up. This is a vertical displacement of 3. Therefore, the translation from shape E to shape F is described as the vector in the image.
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Translations can also occur on sets of axes. This problem can be approached in the same way. Decide on a reference point to help describe the translation.
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From shape V to shape W, the reference point P has moved 5 squares to the right. Check the axes, each square is 1 unit. This is a horizontal displacement of 5
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It then moves 2 squares up. This is a vertical displacement of 2. Therefore, the translation of shape V to shape W is described as the vector in the image above.

Question

What is the translation between shape A and shape B?

Plotting translations

When given a translation, it is possible to plot a shape in its new position.

This can be done by translating each of the shape in turn. These points can then be joined together to create the shape.

Alternatively, since the new shape is , it is possible to pick one vertex as a reference point. If this reference point is moved to the correct position, the whole congruent shape can be drawn.

Examples

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Image caption,
Translate shape A by the vector in the image. Label the new shape B.
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The vector in the image means shape A is translated 6 squares to the left. Starting with reference point P, the vertex is translated 6 squares horizontally to the left.
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Reference point Q is translated 6 squares horizontally to the left.
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Reference point R is translated 6 squares horizontally to the left.
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Finally, reference point S is translated 6 squares horizontally to the left.
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Joining the points up, the new shape is finished. Label the new shape B.
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Shape D is a translation of shape C by the vector in the image. It is possible to draw shape D in its new position. Decide on a single reference point (labelled P). Here, the bottom right corner has been chosen.
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The vector in the image means shape C is translated 3 squares to the right and 2 squares up. Reference point P is translated 3 squares horizontally to the right and 2 squares up.
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Since shape D is congruent (identical) to shape C, an identical triangle can be plotted with the bottom right corner at the new location. Label the new shape D.