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 $y = 14$
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.
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.
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.