# Similar

Two shapes are **Similar** when one can become the other after a **resize**, flip, slide or turn.

## Resizing

If one shape can become another using Resizing (also called** ***dilation, contraction, compression, enlargement* or even *expansion*), then the shapes are **Similar**:

These Shapes are Similar! |
---|

If there is no need to resize, then the shapes are better called Congruent*.

## There may be Turns, Flips or Slides, Too!

Sometimes it can be hard to see if two shapes are Similar, because you may also need to turn, flip or slide a shape.

Rotation | Turn! | |
---|---|---|

Reflection | Flip! | |

Translation | Slide! |

## Examples

Here are 3 examples of shapes that are **Similar:**

Resized | Resized and Reflected | Resized and Rotated |

## Why is it Useful?

When two shapes are similar, then:

- corresponding angles are equal, and
- the lines are in proportion.

This can make life a lot easier when solving geometry puzzles, as in this example:

### Example: What is the missing length here?

Notice that the red triangle has the **same angles** as the blue triangle ...

... they both have one **right angle**, and a **shared angle** in the left corner

In fact we can flip the red triangle over, rotate it a little, resize it, and it will fit exactly on top of the blue triangle. So they are **similar triangles**.

So the line lengths are in proportion:

- The blue triangle has two sides with the ratio 130/127
- The red triangle has
**matching**sides in the ratio ?/80

and we can calculate:

### ? = 80 × \frac{130}{127} = 81.9

(No fancy calculations, just common sense.)

## Congruent or Similar?

Shapes are Congruent when they are the same size (but may have been rotated, reflected or moved). So when the shapes become the same:

When we ... | Then the shapes are ... | |
---|---|---|

... only Rotate, Reflect and/or Translate | ## Congruent | |

... also need to Resize | ## Similar |

## *Are Congruent Shapes also Similar?

Most people (including us) say "**Congruent** shapes are also **Similar**".

### Example:

We can move and rotate the orange shape to exactly match the blue shape, so the two shapes are **Congruent**.

We **don't have to** resize for the shapes to be similar! So they are **also Similar** even though no resizing was needed.