Enlargement - KS3 Maths

Key points

An image of an irregular pentagon enlarged by scale factor three. — Image caption,
Shape A has been enlarged by a scale factor of three. The new shape is shape B.

An is a type of .
An enlargement increases or decreases the size of the shape (). The new shape () is a similar shape.
The increase in size from one shape to another is called a .
The position of the enlarged shape is determined by a point called the .
- If a and the centre of enlargement share the same point, that point will be under the enlargement.
When enlarging shapes, a good understanding of similar shapes and naming and plotting coordinates can be helpful.

Scale factors

When a shape is enlarged, the length of each side is multiplied by the same value. This value is called the .
For example, to enlarge a shape with the scale factor of two, the length of all sides would be multiplied by two. Since the new shape is similar, all of the angles would be the same size.
A scale factor greater than one produces a larger shape. A scale factor between zero and one results in a smaller shape.
When given two shapes, a scale factor can be worked out by dividing corresponding sides.

Examples

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Triangle DEF is an enlargement of triangle ABC.
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The scale factor can be calculated by dividing a pair of corresponding sides. For example, in these shapes EF and BC are corresponding sides. BC has a length of three squares and EF has a length of six squares.
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Scale factor = EF/BC = 6⁄3 = 2. Therefore, triangle DEF is an enlargement of triangle ABC by a scale factor of 2
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The scale factor of triangle ABC compared to triangle DEF can be described in a similar way. This time the sides need to be divided in the opposite order.
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Scale factor = BC/EF = 3⁄6 = 1⁄2. Therefore, triangle ABC is an enlargement of triangle DEF by a scale factor of ½
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Shape B is an enlargement of shape A by a scale factor of three. Therefore, all lengths on shape B need to be drawn three times the size.
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Label the vertices of shape A. In shape A, the vertical line PQ has a length of three squares. Multiply the length of the vertical line by the scale factor of three. 3 × 3 = 9. Shape B has a corresponding side P'Q' with length of nine squares.
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The horizontal line PT has a length of two squares. Multiply the length of the horizontal line by the scale factor of three. 2 × 3 = 6. Shape B has a corresponding side P'T' with a length of six squares.
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The vertical line ST has length of two squares. Multiply the length of the vertical line by the scale factor of three. 2 × 3 = 6. Shape B has a corresponding side S'T' with a length of six squares.
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Continue this process for the final two line segments, QR and RS. The horizontal line QR has length of one square. Multiply the length of QR by the scale factor of three. 1 × 3 = 3. Shape B has a corresponding side Q'R' with a length of three squares. The final side R’S’ can now be drawn to complete the shape. The shape is complete. Shape B is an enlargement of shape A by a scale factor of three.

Question

Shape D is an enlargement of shape C. What is the scale factor that takes shape C to shape D?

An image of a square grid. The grid has a length of fifteen squares and a width of ten squares. Two similar shapes have been drawn on the grid. Shape C, starting from the top right vertex, has sides of length, two squares down, one square to the left, one square up, two squares to the left, one square up and three squares to the right. Shape D, starting from the top right vertex, has sides of length, six squares down, three squares to the left, three squares up, six squares to the left, three squares up and nine squares to the right. Shape C is coloured blue and shape D is coloured orange.

Enlarging by an integer scale factor

Enlarging by a positive scale factor increases the size of the shape. When using a centre of enlargement, the final position of the new shape can be determined.
To work out the position of the image after an enlargement:
1. Pick a on the shape (object).
2. Count the distance between the centre of enlargement and the vertex. This can be broken down into and .
3. Multiply these displacements by the scale factor.
4. Using these values, count from the centre of enlargement to find the position of the corresponding vertex.
5. Repeat the process for additional vertices.
The answer can be checked by drawing lines through corresponding points on the object and image. These will meet at the centre of enlargement.

Examples

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Triangle ABC is to be enlarged by a scale factor of two, using point P as the centre of enlargement.
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Starting with vertex C, the vertex is three squares to the right of the centre of enlargement.
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Multiply this displacement by the scale factor of two. 2 × 3 = 6. The corresponding point C' needs to be six squares to the right of the centre of enlargement.
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Vertex A is three squares to the right and three squares up from the centre of enlargement. Multiply this displacement by the scale factor of two. 3 × 2 = 6. The corresponding point A' needs to be six squares to the right and six squares up from the centre of enlargement.
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Vertex B is five squares to the right and three squares up from the centre of enlargement. Multiply this displacement by the scale factor of two. 5 × 2 = 10 and 3 × 2 = 6. The corresponding point B' needs to be ten squares to the right and six squares up, from the centre of enlargement.
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Draw lines to connect each new vertex to draw the enlarged shape. The enlargement of the shape is complete. Shape A'B'C' is an enlargement of ABC by a scale factor of two, with point P as the centre of enlargement. The answer can be checked by drawing dashed lines through pairs of corresponding points. These lines all meet at the centre of enlargement, point P.
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It is possible to enlarge a shape where the centre of enlargement is inside the shape. Triangle DEF is to be enlarged by a scale factor of three, with point Q as the centre of enlargement.
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Vertex D is two squares up from the centre of enlargement. Multiply this displacement by the scale factor of three. 2 × 3 = 6. The corresponding point D' needs to be six squares up from the centre of enlargement. Vertex E is three squares to the right of the centre of enlargement. Multiply this displacement by the scale factor of three. 3 × 3 = 9. The corresponding point E' needs to be nine squares to the right of the centre of enlargement.
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Vertex F is two squares to the left and two squares down from the centre of enlargement. Multiply this displacement by the scale factor of three. 2 × 3 = 6 and 2 × 3 = 6. The corresponding point F' needs to be six squares to the left and six squares down from the centre of enlargement.
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Draw a line to connect each new vertex. The enlargement of the shape is complete. Shape D'E'F' is an enlargement of DEF by a scale factor of three, with point Q as the centre of enlargement. The answer can be checked by drawing dashed lines through pairs of corresponding points. These lines all meet at the centre of enlargement, point Q.

Question

Shape EFGH is an enlargement of shape ABCD. It has been enlarged using a scale factor of two, with point P as the centre of enlargement.

One vertex on the enlarged shape EFGH is in the wrong position. Which is the incorrect vertex?

An image of a square grid. The grid has a length of fifteen squares and a width of ten squares. Two similar trapeziums have been drawn on the grid. Trapezium A B C D, starting from the top left vertex, A, has sides of length, two squares to the right, four squares down, three squares to the left with the final diagonal joining back to the starting point. Each vertex is labelled A, B, C, and D in a clockwise direction. Trapezium E F G H, starting from the top left vertex, E, has sides of length, four squares to the right, eight squares down, fives squares to the left with the final diagonal joining back to the starting point. Each vertex is labelled E, F, G, and H in a clockwise direction. It has been drawn such that vertex G is two squares below vertex D. A point P has been marked and labelled. P is three squares to the right and two squares below vertex B. Drawn above: An orange arrow pointing to the left. Written above the arrow: scale factor two. Trapezium A B C D is coloured blue. Trapezium E F G H and the arrow are coloured orange.

Enlarging by a fractional scale factor

Enlarging by a decreases the size of the shape.
To enlarge a shape by a fractional scale factor, the same method as enlarging by an integer scale factor can be used.
Multiplying the displacements by a fraction will result in a smaller distance between the centre of enlargement and a corresponding vertex.

Examples

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Shapes can be enlarged on sets of axes. Enlarge triangle ABC by a scale factor ⅓. Use point P with the coordinates of (1, 10) as the centre of enlargement.
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Start with vertex A. The vertex is six squares to the right of the centre of enlargement. Multiply this displacement by the scale factor of a ⅓. 6 × ⅓ = 2. The corresponding point A' needs to be two squares to the right of the centre of enlargement.
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Vertex B is nine squares to the right and six squares below the centre of enlargement. Multiply this displacement by the scale factor of a ⅓. 9 × ⅓ = 3 and 6 × ⅓ = 2. The corresponding point B' needs to be three squares to the right and two squares below the centre of enlargement.
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Vertex C is six squares to the right and six squares below the centre of enlargement. Multiply this displacement by the scale factor of a ⅓. 6 × ⅓ = 2 and 6 × ⅓ = 2. The corresponding point B' needs to be two squares to the right and two squares below the centre of enlargement.
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Draw a line to connect each new vertex. The enlargement of the shape is complete. Shape A'B'C' is an enlargement of a scale factor of ⅓, with point P as the centre of enlargement. The answer can be checked by drawing dashed lines through pairs of corresponding points. These lines all meet at the centre of enlargement, point P.
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It is possible to draw an enlargement by calculating the position of one vertex. The similar shape then needs to be drawn at the correct proportion using the scale factor. Shape A is to be enlarged by a scale factor of ½, with point Q as the centre of enlargement and positioned at the coordinates (3, 11).
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Choose a vertex on the shape. In this example, Vertex X has been chosen with coordinate (1, 5). Vertex X is two squares to the left and six squares below the centre of enlargement. Multiply this displacement by the scale factor of a ½. 2 × ½ = 1 and 6 × ½ = 3. The corresponding point X' needs to be one square to the left and three squares below the centre of enlargement. X' has the coordinates (2, 8).
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Now that X' is in the correct position, a similar shape with edges half the size of B can be drawn. For example, using vertex Y with coordinate (5, 5), the length of XY is four squares long. In the enlargement, multiplying by a scale factor of ½, X'Y' will be two squares long. Similarly, using vertex Z with coordinate (1, 1), the length of XZ is four squares long. In the enlargement, multiplying by a scale factor of ½, X'Z' will be two squares long.
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Continue this process for the remaining edges. The shape is complete. Shape B is an enlargement of shape A by a scale factor of ½. The answer can be checked by drawing dashed lines through pairs of corresponding points. These lines meet at the centre of enlargement, point P.

Question

Triangle PQR is to be enlarged by a scale factor \( \frac{1}{2} \), with point Z as the centre of enlargement and with the coordinates (1, 2).

What is the new coordinate of vertex Q in the enlargement?

The image shows a set of axes. The horizontal axis is labelled x. The values go up in ones from zero to eight. The vertical axis is labelled y. The values go up in ones from zero to eight. Triangle P Q R has been plotted and has vertices with coordinates, P equals five comma seven, Q equals seven comma four, and R equals five comma two. A point Z has been marked and labelled. Z has coordinate one comma two. Triangle P Q R is coloured blue.

Practise enlarging shapes

Quiz

Practise enlarging shapes with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

A female looking at a selection of photographs. — Image caption,
Photographs can be enlarged to produce bigger images.

A professional photographer can produce photos of different sizes. The most common photo size is approximately 4 inches by 6 inches. The length is 1.5 times longer than the width.

Photographs can be enlarged to produce bigger images, such as those that may be used in newspapers or magazines.