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's no need to resize, then the shapes are better called Congruent*.

There may also be Turns, Flips or Slides

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's the missing length here?

A small red right triangle nested inside a larger blue right triangle, sharing one angle

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 ...

The small red triangle flipped and rotated next to the blue triangle

... 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 × 130127 = 81.9

(No fancy calculations, just common sense.)

Example: Are these circles similar?

Two separate circles, Circle A with radius 3 and Circle B with radius 6

First, we slide circle A so its center is exactly on top of circle B's center. Now they are "concentric" (they share the same middle).

Circle A and Circle B sharing the same center point

Now let's resize circle A. The scale factor is 63 = 2, so a simple doubling makes its radius 6:

Circle A resized by a scale factor of 2 to match Circle B perfectly

circle A now perfectly matches circle B

So they are similar! Slide (translation) and resize (dilation) is all it took.

In fact ALL circles are similar:

All circles are the same shape. They don't have different angles or side lengths like triangles or rectangles do
We can always slide one center point onto another
We can always find a scale factor radius Bradius A to change one radius into another (unless the radius is zero, but then it wouldn't be a circle, right?)

Yay for circles!

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) agree that "Congruent shapes are also Similar".

Example:

congruent turn

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.

781, 3272, 6050, 6051, 6052, 3273, 3274, 3275, 631, 632, 1802, 1803