Triangle inequality is an exterior angle of a right triangle and therefore larger than , meaning the other base angle of the isosceles triangle also is greater than and their sum exceeds in violation of the triangle postulate). This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. Consider a triangle whose sides are in an arithmetic progression and let the sides be , , . Then the triangle inequality requires that The first of these quadratic inequalities requires to
Hypotenuse is the square root of 25, that is, 5. The word "hypotenuse" is derived from Greek (sc. or ), meaning "[side] subtending the right angle" (Apollodorus), The Greek term was loaned into Late Latin, as "hypotēnūsa". Adoption as a learned Latinism used in modern languages dates to the 16th century. The spelling in "-e", as "hypotenuse", is French in origin (Estienne de La Roche 1520). The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle (the sides perpendicular to