# Transformations

### 5) Enlargement

Enlarging a shape changes its size.

All the sides of the triangle X’Y’Z’ are twice as long as the sides of the original triangle XYZ. The triangle XYZ has been enlarged by a scale factor of 2.

To enlarge a shape, a centre of enlargement is required. When a shape is enlarged from a centre of enlargement, the distances from the centre to each point are multiplied by the scale factor.

The lengths in triangle A’B’C’ are three times as long as triangle ABC. The distance from O to triangle A’B’C’ is three times the distance from O to ABC.

The triangle ABC has been enlarged by a scale factor of 3 about the centre of enlargement O.Two pieces of information are needed to enlarge a shape:

1) the scale factor
2) the centre of enlargement

##### Positive enlargements

A positive scale factor increases the size of a shape.

Alternatively, the distances OA, OB and OC can be shown as vectors. OA = $\left(\begin{array}{c}3\ 2\end{array}\right)$ so under a scale factor of 2 OA’ = $\left(\begin{array}{c}6\ 4\end{array}\right)$.

The centre of enlargement may be outside an object, or it may be inside a shape, on an edge or at a corner.

The image may overlap the shape or one may be inside the other.

##### Question 1

Enlarge the triangle PQR by a scale factor of 3 about the centre of enlargement O.

Answer: First draw ray lines from O to each corner of the triangle and extend them. Next measure the distance from O to each corner of PQR. Multiply this distance by 3 and plot the points P’ Q’ and R’. Finally join up the points P’ Q’ R’.

##### Question 2

What scale factor has been used to enlarge the shape OXYZ to OX’Y’Z’?

Answer: Each side in the shape OX’Y’Z’ is twice as long as the side in the shape OXYZ. OXYZ has been enlarged by a scale factor of 2 about the centre of enlargement O.

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