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The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse.
The chapter on areas includes both trigonometric formulas and Heron's formula for computing the area of a triangle from its side lengths, and the chapter on inequalities includes the Erdős–Mordell inequality on sums of distances from the sides of a triangle and Weitzenböck's inequality relating the area of a triangle to that of squares on ...
Such triples are necessarily primitive and have the form (2n + 1, 2n 2 + 2n, 2n 2 + 2n +1). This results from Euclid's formula by remarking that the condition implies that the triple is primitive and must verify (m 2 + n 2) - 2mn = 1. This implies (m – n) 2 = 1, and thus m = n + 1.
The Euclidean distance is the prototypical example of the distance in a metric space, [11] and obeys all the defining properties of a metric space: [12] It is symmetric, meaning that for all points and , (,) = (,). That is (unlike road distance with one-way streets) the distance between two points does not depend on which of the two points is ...
Drawing a line connecting the original triangles' top corners creates a 45°–45°–90° triangle between the two, with sides of lengths 2, 2, and (by the Pythagorean theorem) . The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of 3 − 1 {\displaystyle {\sqrt {3}}-1} , 3 + 1 ...
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