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Chapter exercises 2 — Line segment

Exercises and resolutions 6 If the segment $\overline{AB}$ measures 17 cm, determine the value of $x$ in the cases: Ative o JavaScript para ver esta figura interativa. Ver resoluçãoa) $x + 7 = 17 \Rightarrow x = 10$. b) $PA = PB + BA$: $21 = x + 17 \Rightarrow x = 4$. c) $(x + 3) + x = 17 \Rightarrow x = 7$. d) $AP = AB + BP$: $2x = 17 + (x - 3) \Rightarrow x = 14$. Ver respostaa) $x = 10$ cm · b) $x = 4$ cm · c) $x = 7$ cm · d) $x = 14$ cm

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Segments and straight lines in everyday life

Before being objects of theorems, lines and segments were work tools: stretched strings, rulers and plumb lines. Who discovered No one in particular: the first straight lines were drawn by farmers, builders and surveyors from civilizations such as Egypt and Mesopotamia. The Greek texts called harpedonaptai, “rope stretchers”, the Egyptian experts in measuring land; the term appears in a fragment attributed to Democritus (5th century BC). In Rome, surveyors used the groma, a cross with plumb lines, to lay out straight roads and plots.

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Distance between two points

The distance between two points is the length of the segment that joins them — the shortest path between them. Who discovered The idea is intuitive and old; what has an author is its use and its generalization. Around 240 BC, Eratosthenes of Cyrene calculated the distance around the Earth by comparing shadows in Siena and Alexandria. In 1906, the Frenchman Maurice Fréchet realized that three properties were enough — never negative distance, symmetric and obeying the triangular inequality — to talk about “distance” between any objects. In 1914, Felix Hausdorff called this structure metric space.

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Measurement of a segment

Measuring a segment is asking how many times a unit fits in it. The question is simple; the answer took centuries to complete. Who discovered Measuring lengths predates writing, but the person who solved the theoretical problem was Eudoxus of Cnidus (c. 408–355 BC). The Pythagoreans had discovered that the diagonal and the side of a square are incommensurable: there is no unit, no matter how small, that fits an integer number of times in both. Eudoxus created a theory of proportions that works even in this case, recorded in book V of The Elements.

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Midpoint

Finding the exact middle of a segment is one of the oldest and most used constructs in geometry. Who discovered The oldest construction we know is in The Elements by Euclid (c. 300 BC), in proposition I.10: “cut a given finite straight line in half”. Euclid raises a equilateral triangle over the segment, divides the top angle in half and proves, by congruent triangles, that the division falls exactly in the middle. Much later, the Danish Georg Mohr (1672) and the Italian Lorenzo Mascheroni (1797) showed that you can find the midpoint just with a compass, without a ruler.

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Segment congruence

Two segments are congruent when one can be “placed” on top of the other. It’s the geometric way of saying they are the same size — without needing a graduated ruler. Who discovered Euclid dealt with the topic at the beginning of the The Elements (c. 300 BC). His common notion 4 says that “things that coincide with each other are equal”, and the propositions I.2 and I.3 show how to transport a segment and how to cut, from a larger segment, a piece equal to a smaller one. The most curious thing is the tool: Euclides only admits a compass that “closes” when it leaves the paper and, even so, proves that it is possible to copy distances.

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Positions of two segments

Consecutive, collinear, adjacent segments: names that describe how two pieces of straight line fit together. Who discovered This classification does not have an author: it is a modern didactic organization. The practice of joining segments, however, is very old. The Egyptian surveyors, who the Greek historian Herodotus (5th century BC) linked to the origin of geometry, remarked the fields after each flood of the Nile by amending measurements made with ropes. Centuries later, Heron of Alexandria (1st century AD) described in his Dioptra how to measure land using aligned sections.

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Segment and ray

A straight line has no beginning or end. In practice, we almost always work with pieces of it: segments and rays. Who discovered The Greeks thought of the piece first. In The Elements (c. 300 BC), Euclid speaks of “finite straight lines” — our segments — and his second postulate allows a finite straight line to be extended continuously. In other words, for him, the unlimited straight line was something that was obtained by extending a segment as much as necessary. The names “segment” and “ray”, with the definitions used in school, are from modern vocabulary, established by geometry books from the 19th and 20th centuries.

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The notion of "being between"

“B is between A and C.” Euclid used this idea in almost every demonstration—without ever explaining it. Who discovered The first to transform “being between” into axioms was the German Moritz Pasch (1843–1930), professor in Giessen, in the book Lições de Geometria Moderna, by 1882. Before him, Carl Friedrich Gauss had already observed, in a letter from 1832, that words like “between” would need to be clarified in a complete exposition of geometry. David Hilbert incorporated Pasch’s ideas in 1899, in the so-called axioms of order.

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