Geometry

The fourth dimension: drawing what cannot be seen

All the diagrams in this chapter have one advantage: the object represented fits, one way or another, on paper. But mathematics speaks quite naturally of spaces of four, ten or a million dimensions. Algebra has no difficulty at all: a point in four dimensions is just a list of four numbers, $(x,y,z,w)$. The eye, however, lives in three. How do you draw what cannot be seen? Flatland The title page of Flatland: a romance of many dimensions, “by A Square”, in the American edition of 1885. The real author was the English schoolmaster Edwin Abbott.Edwin A. Abbott · Wikimedia Commons · public domain In 1884 Edwin Abbott, headmaster of a school in London, published under the pseudonym A Square a short novel called Flatland. The narrator is a square living in a two-dimensional world inhabited by triangles, squares, pentagons and circles. The inhabitants see only segments: viewed edge-on, every polygon is a line.

October 19, 2026 Read

Sine, leg and hypotenuse: trigonometry and its lost names

Almost every student learns to say “sine”, “cosine”, “leg” (or cathetus) and “hypotenuse” without knowing what these words mean. They are labels to memorise, like street names in a strange city. But each of them was once a description, and some hide unexpected stories. The sine, for example, is the result of a translation that went wrong. Pythagoras in Latin The most famous theorem of geometry was for a long time a sentence about real squares, drawn on the sides of a triangle. The old texts on this site wrote it in the medieval manner:

October 9, 2026 Read

Euclid: the figure and the proof

Around 300 BC, in Alexandria, Euclid gathered Greek geometry into thirteen books, the Elements. For more than two thousand years it was the most widely read mathematics book in the world, and its first proposition shows how it works: On a given finite straight line, to construct an equilateral triangle. — Euclid, Elements, book I, proposition 1 The recipe is short. Let $AB$ be the given line. With centre $A$ and radius $AB$, draw a circle; with centre $B$ and the same radius, draw another. Call $C$ a point where the two circles cut each other. Then $AC=AB$, because both are radii of the first circle, and $BC=AB$, because both are radii of the second. Things equal to the same thing are equal to each other, so $AC=BC=AB$, and triangle $ABC$ is equilateral.

October 9, 2026 Read

FUVEST 2022 – 1st phase: Mathematics solved

Source: FUVEST – General Knowledge test, version V and rectified official template of the 1st phase (applied in 12/12/2021). The 1st phase of FUVEST 2022 had 90 questions from all subjects. This post brings the 12 mathematics questions, which in test V are questions 30 to 41. In the others versions (K, Q, X and Z) the same questions appear with different numbering and another order of alternatives, so always check the version of your test.

September 27, 2026 Read

Right Triangles

A triangle is a three-sided polygon, in which we use straight segments for the sides. It is a particular case of quadrilaterals, as it is a quadrilateral divided by 2. Vertices: can be defined as meeting points between the sides of a triangle. Generally speaking, vertices are meeting points between sides of any polygon. In particular, a triangle is formed by 3 vertices. Diagonal: is the straight line segment that connects two non-consecutive vertices of any flat and spatial polygon. One of the important properties of triangles is that they have diagonals. An important observation is that there is no triangle that has two non-consecutive vertices. The diagonal of a quadrilateral, for example, is one of the sides of a triangle.

May 4, 2024 Read

Angles in basic planes

Angles are key pieces for the development of geometry. It is present in our daily lives such as stairs, building cranes, constructions such as the Leaning Tower of Pisa, among others. In Euclid’s book Elements, this concept was well worked on, even decades before Christ. Examples of angle applications Next, we present some applications of angles. Building Crane Tower of Pisa Ladder A widely used example is the clock, with the movement of the hour and minute hands, the angles are formed and the time is defined.

August 28, 2022 Read

Property of Angles in Planes

In the previous post we talked about several important definitions for angles in flat figures. Now, we deepen the concepts through the properties of angles for applications in flat figures. angle bisector of an angle Definition: the angle bisector of an angle is a ray, which starts from the vertex of the angle and divides it into two congruent angles ($\cong$). angle bisector produces two congruent angles. Ampliar figura para ler os detalhes From the figure, we notice that

August 28, 2022 Read