Similarity and Congruence

Two objects are similar if they have the same shape, so that one is an enlargement of the other.

Two objects are congruent if they are the same shape and size.

Scale Models and Factors

In an enlargement or reduction

· Corresponding angles are equal

· Corresponding lengths are in the same ratio - they have the same Scale Factor

· The scale factor is greater than 1 for an enlargement

· The scale factor is smaller than 1 for a reduction

Triangle A is an enlargement of C

with a scale factor of 2

Triangle C is a reduction of A with a scale factor of

Triangle B is an enlargement of C

with a scale factor of 3

Triangle B is an enlargement of A

with a scale factor of 3 :2

Its scale factor may be written as

Two shapes are said to be similar if one is a scaled version of the other.

This means that

i. their corresponding angles are equal

ii. their corresponding sides are in the same ratio. ( ie. they have the same scale factor)

Are these dominoes similar?

i. Check that their corresponding angles are equal.

(All right angles, so equal)

ii. Check the ratio of their corresponding sides.

Rectangular Callout: Scale factor
1.5 or 3:2

The dominoes are similar

Calculate the size of side x.

For any triangle

If pairs of corresponding angles are equal, then ratios of lengths of corresponding sides are equal, and the triangles are similar.

Conversely, if ratios of lengths of corresponding sides are equal, then pairs of corresponding angles are equal , and the triangles are similar.

Example

Find the length of BC

BC is parallel to DE , so the corresponding angles are equal.

This means that triangles ABC and ADE are similar.

Split the triangles into their components