Proportionality

Exercises from chapter 13 — Similarity of triangles and point powers

Exercises and resolutions 445 The triangles $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$ in the figure are similar. If the similarity ratio of the 1st to the 2nd is $\dfrac{3}{2}$, determine: a) $a$, $b$ and $c$; b) the ratio between their perimeters. Ver respostaa) $a = 21$, $b = 18$ e $c = 15$ · b) $\dfrac{3}{2}$ 446 Triangles $ABC$ and $PQR$ are similar. Determine $x$ and $y$. Ver resposta$x = 16$ e $y = 14$

October 11, 2026 Read

Point power

Two chords that intersect within a circle: the product of the parts of one is equal to the product of the parts of the other. With this idea, the distance to the horizon can be calculated. Who discovered The two classical forms are in The Elements of Euclid (c. 300 BC): the proposition III.35, about cords that cut each other, and the III.36, on a secant and a tangent drawn from the same exterior point. The name power of a point was created much later, in 1826, by the Swiss Jakob Steiner, who realized that the product is always the same, whatever the straight line drawn by the point.

October 11, 2026 Read

Cases (criteria) of similarity

To ensure that two triangles are similar, it is not necessary to check everything: two angles, or three ratios between sides, are enough. Who discovered The three criteria are in book VI of The Elements by Euclid (c. 300 BC): triangles equiangular have proportional sides (VI.4, the case AA), triangles with three proportional sides are similar (VI.5, LLL), and two proportional sides with the angle between them equal are also sufficient (VI.6, LAL). Much earlier, tradition attributes to Tales the use of triangles of the same shape to measure inaccessible distances.

October 11, 2026 Read

Similarity of triangles

Two figures are similar when they have the same shape, even if they are different sizes. It’s the mathematics of photos, maps and models. Who discovered The idea is in the stories about Tales of Miletus, who would have measured the height of a pyramid by comparing its shadow with that of a stick. The complete theory is in book VI of The Elements by Euclid (c. 300 BC), which defines similar figures as those with equal angles and proportional sides. It depends on the theory of proportions by Eudoxus of Cnidus (4th century BC), from book V, which also allowed the treatment of ratios between incommensurable segments.

October 11, 2026 Read

Reading: Legendre: towards a rigorous and didactic geometry

Further reading Leitura Legendre: towards a rigorous and didactic geometry In ancient Greece, axiom seemed to mean a truth common to all fields of study, and postulate, a truth of a specific field. Euclid assumed five postulates — among them, that from any point a straight line is drawn to any other, that every limited straight line extends, that with any center and radius a circle is drawn, that right angles are equal, and the famous fifth, that of parallels — and five axioms, such as “things equal to the same thing are equal to each other” and “the whole is greater than the part”. From then on, The Elements rattle off theorems without exercises or applications: a text that is not very didactic by modern standards. Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain In 18th-century France several books appeared to make Euclid more accessible, but many sacrificed rigor. A notable exception was Elements of Geometry by Adrien-Marie Legendre (1752–1833). Legendre shined in his research — in elliptic functions and number theory, in which he completely enunciated the law of quadratic reciprocity — and his geometry book, which reconciled rigor and didactic, had twenty editions in France during the author’s lifetime. In 1809 it was translated in Rio de Janeiro and adopted at the Royal Military Academy, created the following year. For about forty years, Legendre tried to prove the fifth postulate from the others — an impossible task, as we now know — but even in these attempts he showed great ingenuity. Adrien-Marie Legendre, caricature by Julien-Léopold Boilly.Julien-Léopold Boilly · Wikimedia Commons · Public domain

October 11, 2026 Read

Exercises from chapter 12 — Thales' Theorem

Exercises and resolutions 413 Determine the value of $x$ in each case below, with $r$, $s$ and $t$ parallel lines. Ver respostaa) 3 · b) 12 · c) 15 · d) 6 414 In the figures, the lines $r$, $s$ and $t$ are parallel. Determine the values of $x$ and $y$. Ver respostaa) $x = \dfrac{10}{3}$ · b) $x = \dfrac{25}{6}$ · c) $x = \dfrac{10}{3}$ e $y = \dfrac{18}{5}$

October 11, 2026 Read

angle bisectors Theorems

The angle bisector of an angle of a triangle divides the opposite side into parts proportional to the other two sides. Who discovered The internal angle bisector theorem is the proposition VI.3 of the The Elements of Euclid (c. 300 BC). The one with external angle bisector is not in the Greek text: it was added as “proposition A” of book VI by the Scotsman Robert Simson, in his edition of The Elements of 1756, which served as the basis for teaching geometry in the United Kingdom for more than a century. The two angle bisectors are linked to the circle of Apollonius, studied by Apollonius of Perga (3rd century BC).

October 11, 2026 Read

Thales' theorem

A bundle of parallel lines cuts two transversals into proportional parts. It is with this idea that a board is divided into equal parts without doing any math. Who discovered The result is called Tales of Miletus (6th century BC), associated by tradition with the measurement of the height of the Great Pyramid by shadows (see the curiosity about it at the opening of the chapter). The demonstration we know is from Euclid, in proposition VI.2 of The Elements (c. 300 BC), and uses areas of triangles. The name “Tales’ theorem” for this result is a tradition in countries such as Brazil, Portugal, France, Spain and Italy; in German and English, the same name refers to the angle inscribed in the semicircle.

October 11, 2026 Read

ENEM 2024 – Solved mathematics: questions 151 to 165

Source: INEP – proof and template of the 2nd day, Notebook 5 – Yellow. The ENEM Mathematics and Technologies test 2024 was administered on the 2nd day and will from question 136 to 180. This post addresses questions 151 to 165. The numbering used here is that of the Notebook 5 – Yellow; us In other notebooks the same questions appear in a different order. Template Question Subject Template 151 Cylinder volume D 152 Affine function: trend line B 153 Reading Charts: Tank Filling B 154 Statistics: arithmetic mean C 155 Circular sector area E 156 Related function: price modeling B 157 Flow and percentage D 158 Proportional quantities: units C 159 Cost optimization C 160 1st degree equation C 161 Bar chart and percentage A 162 Orthogonal projection C 163 Scales and magnification A 164 Arithmetic mean B 165 Fundamental counting principle B Question 151: Larger volume of cylindrical packaging Statement An industry forms a partnership with a juice distributor to launch market two types of packaging. To manufacture these packaging, the industry has rectangular aluminum sheets, with dimensions 10 cm per 20 cm. Each of these sheets is used to form the lateral surface of the packaging, in the shape of a straight circular cylinder, which subsequently receives circular bottom and lid. The figure illustrates, depending on which of the two extensions will be used as height, the two options to form the possible packaging.

September 26, 2026 Read