Metric geometry: proportionality, similarity and scales
This block supplies the numerical backbone of the paper: it converts real dimensions into drawing dimensions and solves proportional problems by construction rather than by arithmetic. Expect a construction of a proportional segment or a graphic scale, sometimes applied to a plan or a map.
Proportionality constructions
- Thales: parallel lines cut two transversals in proportional segments; this divides a segment into n equal parts and finds the fourth proportional.
- Third proportional to a and b: x = b²/a, obtained by the right-angle construction with the two given segments on the same line.
- Mean proportional (geometric mean) to a and b: x = √(a·b), the altitude of a right triangle whose hypotenuse is a + b, or the leg with the semicircle of Thales.
- Golden section of AB: point P with AB/AP = AP/PB = φ ≈ 1.618. Erect a perpendicular at B of length AB/2, join the other end to A, subtract that half-side from the hypotenuse and carry the remainder onto AB from A.
- Golden rectangle: the square removed from it leaves a similar rectangle; adding squares around a common corner produces the golden spiral.
Similarity and homothety
Figures with ratio k have areas in ratio k² and volumes in k³. A homothety of centre O and ratio k places A′ on OA with OA′ = k·OA; it is the tool for enlarging a figure and for locating circles tangent to two given circles.
Scales
Scale E = drawing / reality. 1:50 means the drawing is one fiftieth of reality; 2:1 doubles. To build a graphic (decimal) scale: choose a base representing a round real measure, divide it into equal parts, and add a counter-scale to the left with the first part subdivided into ten for reading decimals. A scale of transversals refines this to hundredths. The escalímetro (triangle of scales) reads a fixed set of scales directly. Areas scale with k² and volumes with k³, so a plan at 1:50 shows areas 2 500 times smaller.
Worked example
A wall 4.5 m long at scale 1:75 measures 4500/75 = 60 mm. To build a decimal scale for 1:75: one metre is 13.33 mm, so take a base of 15 m = 200 mm, divide it into three parts of 5 m each, and subdivide the first part into five for single metres.
Traps
Reading the counter-scale in the wrong direction (it grows towards the left, away from zero); applying k instead of k² to areas; measuring a photocopied drawing with a ruler instead of the graphic scale; writing 1:50 when the drawing is larger than reality, where the scale is 50:1.