Transformations, congruence, vectors and similarity Study Guide

Enlargement is a transformation that changes the size of an object, but not its shape. It requires two key pieces of information: a centre of enlargement and a scale factor. The centre of enlargement is a fixed point from which all points on the object are scaled. The scale factor determines how much larger (or smaller) the image will be compared to the original object.

A positive scale factor means the image is on the same side of the centre of enlargement as the object. If the scale factor is greater than 1, the image is larger. If it's between 0 and 1, the image is smaller. A negative scale factor means the image is on the opposite side of the centre of enlargement from the object. If the absolute value of the scale factor is greater than 1, the image is larger. If the absolute value is between 0 and 1, the image is smaller.

To enlarge a shape:

1. Draw a line from the centre of enlargement through each vertex of the object.
2. Measure the distance from the centre of enlargement to each vertex.
3. Multiply these distances by the scale factor.
4. Measure the new distances along the lines drawn in step 1, from the centre of enlargement. Mark the new vertices.
5. Connect the new vertices to form the enlarged image.

For negative scale factors, the process is similar, but you extend the lines from the centre *through* the object and then measure the scaled distance on the *opposite* side of the centre. This effectively rotates the object by 180 degrees around the centre of enlargement as well as scaling it.

Similar shapes are shapes that have the same form but are not necessarily the same size. This means that one shape is an enlargement or reduction of the other. The key properties of similar shapes are that all corresponding angles are equal, and all corresponding sides are in the same proportion. The ratio of corresponding sides is called the scale factor. For example, if two triangles are similar, their internal angles will match up perfectly, even if one triangle is much bigger than the other. If one side of the larger triangle is twice as long as the corresponding side of the smaller triangle, then all other corresponding sides will also be twice as long. This scale factor can be used to find unknown side lengths in similar shapes. To calculate the scale factor, divide the length of a side in the larger shape by the length of the corresponding side in the smaller shape. If you are going from a smaller shape to a larger shape, you multiply by the scale factor. If you are going from a larger shape to a smaller shape, you divide by the scale factor. This concept is fundamental in many areas of mathematics and geometry, including map interpretation and architectural scaling.