10_Geometry — how few numbers fix a shape?¶
Level: 101 · for anyone reviewing geometry before precalculus, or coming back to it after many years
The geometry every precalculus book assumes, in the order Sullivan's review appendix gives it (Section A.2, Geometry Essentials): the Pythagorean theorem and its converse, the formulas for area and volume, and congruent and similar triangles. It is a prerequisite chapter, meant to be read just in time: the first section of the book proper, the distance formula, opens by sending the reader here. Each lesson ends with the book's questions and some of its own, answers folded away, and a deck of Anki flashcards whose trap cards are built around the tempting wrong answer.
| # | Lesson | The question it answers |
|---|---|---|
| 1 | The Pythagorean theorem and its converse | When do three lengths make a right angle, and why must the longest side go on the left? |
| 2 | Area and volume formulas | How do you tell eleven formulas apart without memorising them, and catch a wrong one before using it? |
| 3 | Congruent and similar triangles | Which three measurements fix a triangle, which fix only its shape, and why is SSA missing? |
The through-line¶
A shape is decided by a few lengths, and everything else follows from them. Lesson 1: in a right triangle, two sides decide the third, and three sides decide whether the angle is right at all. Lesson 2: an area is two lengths multiplied and a volume three, so scaling every length by k scales area by k² and volume by k³, a test that catches a misremembered formula. Lesson 3: three well-chosen measurements decide a whole triangle, while three angles decide only its shape, and the shapes that match are the similar ones, related by one scale factor k, which is lesson 2's k again. The distance formula then puts lesson 1 into coordinates, and every later page of the book measures with it.
Where this is taught¶
Every classroom precalculus book has a geometry review: Sullivan's Appendix A.2, Stewart's and Larson's appendices, Blitzer's inside covers. For which book to read, and when to read its review appendix, see Precalculus: a reading guide.
Po polsku, w skrócie¶
Rozdział z geometrii, którą zakłada każdy podręcznik do precalculusu: twierdzenie Pitagorasa i twierdzenie do niego odwrotne, wzory na pola i objętości, trójkąty przystające i podobne. Wspólna myśl: kształt wyznacza kilka długości, a reszta z nich wynika. Dwa boki trójkąta prostokątnego wyznaczają trzeci; pole to dwie długości pomnożone, objętość trzy, więc przy skali k pole rośnie k², a objętość k³; trzy dobrze wybrane pomiary wyznaczają cały trójkąt, a trzy kąty tylko jego kształt. To rozdział „do przeczytania w porę": wraca się do niego, gdy wzór na odległość w rozdziale 1 podręcznika tego wymaga. Powrót do podstaw nie jest porażką, tylko zwykłą drogą każdego, kto uczy się matematyki.
Auf Deutsch: Stichwörter¶
Wie wenige Zahlen eine Figur festlegen: Pythagoras und seine Umkehrung, Flächen- und Volumenformeln, kongruente und ähnliche Dreiecke.
Stichwörter: Geometrie, Satz des Pythagoras, Fläche, Volumen, Dimension, Kongruenz, Ähnlichkeit, Dreieck.
A note on the code¶
Lengths are whole numbers or fractions wherever they can be, so "equal" in the Pythagorean test means equal, and answers with π are carried exactly as a + bπ until the last step. Only the construction of a triangle from an angle, in lesson 3, needs sines and cosines, and there the output is rounded to two decimals.