Exercises and resolutions 189 If $\overline{AH}$ is height relative to the $\overline{BC}$ side of $\triangle ABC$, determine $\hat{B}$ and $\hat{C}$ in the cases: Ver respostaa) $\hat{B} = 70^{\circ}$ e $\hat{C} = 40^{\circ}$ · b) $\hat{B} = 60^{\circ}$ e $\hat{C} = 40^{\circ}$ 190 In each case below we have a base isosceles triangle $\overline{BC}$. Determine the angle of the base. Ver respostaa) $65^{\circ}$ · b) $61^{\circ}$
The shadow of an object with the Sun overhead is an orthogonal projection. Engineers turned this idea into a drawing method still used today. Who discovered The idea of projecting figures onto a plane is old — sundials already read the projected shadow of the gnomon. The person who transformed it into a method was the Frenchman Gaspard Monge (1746–1818), creator of descriptive geometry. Monge represented objects by their orthogonal projections on two perpendicular planes, the plan and the elevation, and thus solved construction problems on paper. The distance from a point to a straight line as the length of the perpendicular is a classic result of The Elements of Euclid.
From a point outside a straight line, how can you lower a perpendicular to it? This is one of the few old issues with a registered author name. Who discovered Proclus attributes to Enopides of Chios (5th century BC) the first study of the problem of drawing a perpendicular to a straight line through a point outside it; Oenopides called this line “gnomonic”, in reference to the gnomon, the vertical rod of sundials. The classic solutions are in The Elements of Euclid: proposition I.11 raises the perpendicular by a point on the line, and I.12 lowers it by a point outside it.
Walls, squares, sheets of paper: the built world is full of right angles. Who discovered The right angle was used in practice by Egyptians, Babylonians, Indians and Chinese long before it was defined. The Babylonian tablet Plimpton 322 (c. 1800 B.C.) lists sets of numbers that form right triangles, such as 45, 60, and 75 — the triangle 3-4-5 magnified 15 times —, and the Chinese text Zhoubi Suanjing brings a famous figure about the triangle 3-4-5. The rigorous definition is from Euclid (definition I.10): when a straight line meets another forming equal adjacent angles, each of them is straight, and the straight line is perpendicular to the other.