Editing Right triangle (section)

{{short description|Triangle containing a 90-degree angle}}
[[File:Rtriangle.svg|right|thumb|upright=1.25|A right triangle {{math|△''ABC''}} with its right angle at {{mvar|C}}, hypotenuse {{mvar|c}}, and legs {{mvar|a}} and {{mvar|b}},]]
A '''right triangle''' or '''right-angled triangle''', sometimes called an '''orthogonal triangle''' or '''rectangular triangle''', is a [[triangle]] in which two [[Edge (geometry)|sides]] are [[perpendicular]], forming a [[right angle]] ({{frac|1|4}} [[turn (unit)|turn]] or 90 [[degree (angle)|degrees]]).

The side opposite to the right angle is called the ''[[hypotenuse]]'' (side <math>c</math> in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: ''[[cathetus]]''). Side <math>a</math> may be identified as the side ''adjacent'' to angle <math>B</math> and ''opposite'' (or ''opposed to'') angle <math>A,</math> while side <math>b</math> is the side adjacent to angle <math>A</math> and opposite angle <math>B.</math>

Every right triangle is half of a [[rectangle]] which has been divided along its [[diagonal]]. When the rectangle is a [[square]], its right-triangular half is [[isosceles triangle|isosceles]], with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is [[scalene triangle|scalene]].

Every triangle whose [[base (geometry)|base]] is the [[diameter]] of a [[circle]] and whose [[apex (geometry)|apex]] lies on the circle is a right triangle, with the right angle at the apex and the hypotenuse as the base; conversely, the [[circumcircle]] of any right triangle has the hypotenuse as its diameter. This is [[Thales' theorem]].

The legs and hypotenuse of a right triangle satisfy the [[Pythagorean theorem]]: the sum of the areas of the squares on two legs is the area of the square on the hypotenuse, <math>a^2 + b^2 = c^2.</math> If the lengths of all three sides of a right triangle are integers, the triangle is called a '''Pythagorean triangle''' and its side lengths are collectively known as a ''[[Pythagorean triple]]''.

The relations between the sides and angles of a right triangle provides one way of defining and understanding [[trigonometry]], the study of the metrical relationships between lengths and angles.