Question

Mathematics and Its History - Stilwell
2.3 Ruler and Compass Constructions

2.3.2 Use similar triangles to explain why vi is the length shown in Figure 2.5, and hence show that vI is constructible from l. One of the finest ruler and compass constructions from ancient times is that of the regular pentagon, which includes, yet again, the golden ratio τ = (1 + V5)/2.

Answer 1

Given that BC = l & CD = 1 ACD & ACB are right angled triangle and are similar to each other We can use SAS property of similar triangle to prove the similarity of the triangle < ACB = < ACD (Both 90 degrees).CD/ AC = AC/ CB AC^2 = (CB)CD AC = squareroot (CB) (CD) CD = 1 CB = l AC = squareroot (l) (1) AC = squareroot l