**Similarity**

Two shapes are similar if they are obtained by enlarging, reducing, inverting or mirroring each other.

The shapes above are similar. In similar shapes, the ratio of corresponding sides are maintained.

If the length of one side is halved, the length of every side is halved. If the length of one side is doubled, the length of every side is doubled. The ratio of the corresponding (lengths) is called the scale and is denoted by the letter* k.*

**Example 1**

The shapes below are similar. Find the length of side *b*.

Because the shapes are similar, the ratios of corresponding sides are maintained.

### Similarity of triangles

Triangles are similar if and only if corresponding angles are equal.

**Example 2**

Find the length of side *x*

The triangles are similar because they have two equal angles. The ratio of corresponding sides are maintained.

### Scale

Maps are similar with nature and with other maps too. Maps show a scale that is a ratio of lengths. If a scale is *1: 50 000*, this means that *1 cm* measured from the map, corresponds to a natural length of *50 000* *cm*.

**Example 3**

The scale of a map is *1: 20 000. *A length of *10 cm* is measured on the map. What is the corresponding length in nature?

We mark the length in nature with the letter* x*.

In nature the length corresponds to *200 000 cm*, which is* 2 km.*

### Area and volume

The ratio of areas is the scale squared and the ratio of volumes is the scale to the power of* 3*.

Scale means the ratio of lengths.

**Example 4**

The shapes below are similar. What is the area of the smaller shape?

The ratio of areas is the ratio of lengths (scale) squared. We mark the area of the smaller shape *Ap.*

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