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Exercises from chapter IX — Right triangles

Always draw a figure: mark the right angle, the data and the unknown. Then choose the right triangle in which the unknown and the data appear together. Metric relationships 321 Calculate the elements $a$, $h$, $m$ and $n$ indicated in the figure (categories $b = 4$ and $c = 3$). Ative o JavaScript para ver esta figura interativa. See the resolutionBeing $c = 3$ and $b = 4$: $$a^2 = b^2 + c^2 = 16 + 9 = 25 \Rightarrow a = 5$$$$h = \frac{bc}{a} = \frac{4 \cdot 3}{5} = \frac{12}{5}$$$$m = \frac{c^2}{a} = \frac{9}{5}$$$$n = \frac{b^2}{a} = \frac{16}{5}.$$

October 24, 2026 Read
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Solving right-angled triangles

Solving a right-angled triangle means calculating its main elements: the acute angles ($\hat{B}$ and $\hat{C}$) and the sides ($a$, $b$, $c$). To do this, it is necessary to provide two information about the triangle, at least one of which is the measurement of a segment connected to it (side, height, median, angle bisector etc.) — only with angles, we know the shape but not the size. The story Solving right triangles was, for centuries, the daily work of surveyors, navigators and artillerymen. To measure angles on the ground, the Englishman Leonard Digges described an instrument he called theodelitus — the ancestor of the theodolite, which to this day appears on the streets on a tripod. The description appeared in Pantometria (“Universal Measurement”), a book that Leonard left unfinished and that his son, Thomas Digges, completed and published in 1571. With a measured angle and a known distance, the trigonometric tables did the rest.

October 24, 2026 Read
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Trigonometric ratios in the right triangle

Let’s prove three properties that relate the measurements of the sides to the measurements of the angles of a right-angled triangle. They link the trigonometry of the cycle, which we have studied so far, to the trigonometry of triangles, where it all began. The story It is said that Tales of Miletus (6th century BC) measured the height of the Great Pyramid of Egypt by comparing its shadow with the shadow of a stick: at the same instant, the Sun forms the same angle with both objects, and the ratio between height and shadow is the same for both. This is exactly the idea of ​​this chapter — once an angle is fixed, the ratio between two sides of the right triangle is fixed — although, for Thales, there were still no names or tables for these ratios. The story came to us from much later authors, such as Diogenes Laertius and Plutarch.

October 24, 2026 Read
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Metric elements and relationships

We know that a triangle is rectangle when one of its internal angles is right. In this chapter we will develop a geometric-trigonometric theory that allows us to calculate the measurements of segments and angles of a right-angled triangle based on a minimum number of data. The story The metric relationships of the right triangle are among the oldest in mathematics. Babylonian tablets from around 1800 BC, such as the famous Plimpton 322, list suits of numbers that satisfy $a^2 = b^2 + c^2$, and the YBC 7289 tablet gives an excellent approximation of $\sqrt{2}$, the diagonal of the square on its side 1. The first known proof of the theorem, however, is Greek: it appears as the proposition 47 of book I of Euclid’s The Elements (around 300 BC), and tradition attributes it to the school of Pythagoras.

October 24, 2026 Read
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Chapter exercises 14 — Right triangles

Metric relationships 506 Determine the value of $x$ in the following cases: Ver respostaa) 5 · b) 12 · c) $\sqrt{7}$ · d) $3\sqrt{3}$ 507 Determine $x$ as a function of $a$ in the following cases: Ver respostaa) $a\sqrt{3}$ · b) $\dfrac{a}{2}$ 508 Determine $x$ in cases: Ver respostaa) 12 · b) 24 · c) 20 · d) 8

October 9, 2026 Read
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Applications of the Pythagorean theorem

The diagonal of the square, the height of equilateral triangle and the trigonometric ratios of 30°, 45° and 60° all come from the same theorem. Who discovered The diagonal of the square has already been calculated by Babylonians: tablet YBC 7289 (c. 1800–1600 B.C.) gives $\sqrt{2}$ with six correct decimal places. The ratios between the sides of a right triangle began as astronomical tables: Hipparchus of Nicaea (2nd century BC) set up a string table, the Indian Aryabhata (499) started to use the half-rope — our sine — and Islamic astronomers like Abu al-Wafa (10th century) organized shadow tables, our tangents. The formula that generates all Pythagorean triangles, with sides $x^2 - y^2$, $2xy$ and $x^2 + y^2$, appears in a motto from book Elements of Euclid.

October 9, 2026 Read
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Metric relationships

The square of the hypotenuse is the sum of the squares of the legs. But this is just one of several relationships that height reveals within the right triangle. Who discovered The relationship between the sides of the right triangle is much older than Pythagoras of Samos (6th century BC). Tablets from Babylon from around 1800 BC use terms like 3, 4, 5 and calculate the diagonal of the square; the Sulba Sutras of India (c. 800–500 B.C.) enunciate the rule for building altars; and the Chinese text Zhoubi Suanjing provides visual proof. Greek tradition associated the theorem with Pythagoras, but there is no documentary proof that he demonstrated it. The oldest demonstration that has come down to us is in the The Elements of Euclid (c. 300 BC): the proposition I.47 proves the theorem by areas, and the I.48 proves the reciprocal. The relations with height — each leg and height as proportional averages — are in the proposition VI.8 and its corollary.

October 9, 2026 Read