Chapter 5 of seven · Geometry & Algebra
The bridge
Similarity, and coordinates. Meets at the distance formula being Pythagoras.
- Similar figures; scale factorSame shape, any size — and the one number that says how much bigger.
- Two equal angles are enough (AA)You never have to measure a single side to know two triangles are similar.
- Corresponding sides are in constant ratioOne known pair of sides is enough to predict every other side, exactly.
- Area scales by the square of the scale factorDouble every length and the area does not double — it quadruples.
- The intercept theoremTwo rays from a point, cut by a pair of parallel lines, split into matching ratios.
- The plane as pairs of numbersA point is not just a mark on a page — it is exactly one pair of numbers.
- Distance between two points is PythagorasThe gap between two coordinates is always a right triangle in disguise.
- MidpointThe point exactly between two others is just the average of their coordinates.
- Drawing a relationshipA rule connecting x and y is not a number — it is a whole line of points at once.