S2 · Geometry · GEO.SIMILARITY

Similarity and scale

Intuition

Understand first, compress later.

Geometry links shape, length, angle, area and space. Sketches make relationships visible.

Switch representations

Pythagoras' theorema = 3, b = 4, c = 5

square on leg asquare on leg bsquare on the hypotenuse cright angle

The drawing is to scale: the legs measure 3 and 4, the hypotenuse exactly 5. The two squares on the legs together cover exactly as much area as the square on the hypotenuse: 9 + 16 = 25. Count the unit squares — they come out exactly even. This holds only because the angle between the legs is a right angle; the small square in the corner says exactly that.

Formal view

You can similarity and scale when you can recognise the underlying objects, select valid rules and check each step for equivalence or logical implication.

Why does it work?

This skill sits on a prerequisite network. If a step is unstable, repair the missing prerequisite instead of merely doing more items of the same type.