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
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.
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.
The Pythagorean Theorem is one of the best-known and fundamental propositions in mathematics, establishing an elegant relationship between the sides of a right-angled triangle. The history of this theorem is as fascinating as its mathematical application, as it spans cultures and eras, revealing how mathematical knowledge is accumulated and shared across generations. Traditionally attributed to the Greek philosopher and mathematician Pythagoras, who lived between approximately 570 BC and 495 BC, the theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, known as legs.