similar triangles

Two shapes are Similar when the only difference is size (and possibly the need to move, turn or flip one around).

Resizing is the Key

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

resize These Shapes are Similar!

There may be Turns, Flips or Slides, Too!

Sometimes it can be hard to see if two shapes are Similar, because you may need to turn, flip or slide one shape as well as resizing it.

Rotation rotation 2d Turn!
Reflection reflect on graph Flip!
Translation translation on graph Slide!


These pairs of shapes are all Similar:

similar resize similar resize flip similar resize rotate
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?

similar triangles


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 × 130127 = 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  right arrow


... also need to Resize right arrow


Are Congruent Shapes also Similar?

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


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.