Right triangle - wikipedia Right triangle Right triangle A right triangle ( American English ) or right - angled triangle ( British English ) is a triangle in which one angle is a right angle ( that is, a 90 - degree angle ). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse ( side c in the figure ). The sides adjacent to the right angle are called legs ( or catheti, singular : cathetus ). Side a may be identified as the side adjacent to angle B and opposed to ( or opposite ) angle A, while side b is the side adjacent to angle A and opposed to angle B. If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. Contents 1 Principal properties 1.1 Area 1.2 Altitudes 1.3 Pythagorean theorem 1.4 Inradius and circumradius 2 Characterizations 2.1 Sides and semiperimeter 2.2 Angles 2.3 Area 2.4 Inradius and exradii 2.5 Altitude and medians 2.6 Circumcircle and incircle 3 Trigonometric ratios 4 Special right triangles 4.1 Kepler triangle 5 Thales ' theorem 6 Medians 7 Euler line 8 Inequalities 9 Other properties 10 See also 11 References 12 External links Principal properties Area As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area T is T = 1 2 a b ( displaystyle T = ( \tfrac ( 1 ) ( 2 ) ) ab ) where a and b are the legs of the triangle. If the incircle is tangent to the hypotenuse AB at point P, then denoting the semi-perimeter ( a + b + c ) / 2 as s, we have PA = s − a and PB = s − b, and the area is given by T = PA ⋅ PB = ( s − a ) ( s − b ). ( displaystyle T = ( \text ( PA ) ) cdot ( \text ( PB ) ) = ( s-a ) ( s-b ). ) This formula only applies to right triangles. Altitudes Altitude of a right triangle If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other. From this : The altitude to the hypotenuse is the geometric mean ( mean proportional ) of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. In equations, f 2 = d e, ( displaystyle displaystyle f ^ ( 2 ) = de, ) ( this is sometimes known as the right triangle altitude theorem ) b 2 = c e, ( displaystyle displaystyle b ^ ( 2 ) = ce, ) a 2 = c d ( displaystyle displaystyle a ^ ( 2 ) = cd ) where a, b, c, d, e, f are as shown in the diagram. Thus f = a b c. ( displaystyle f = ( \frac ( ab ) ( c ) ). ) Moreover, the altitude to the hypotenuse is related to the legs of the right triangle by 1 a 2 + 1 b 2 = 1 f 2. ( displaystyle ( \frac ( 1 ) ( a ^ ( 2 ) ) ) + ( \frac ( 1 ) ( b ^ ( 2 ) ) ) = ( \frac ( 1 ) ( f ^ ( 2 ) ) ). ) For solutions of this equation in integer values of a, b, f, and c, see here. The altitude from either leg coincides with the other leg. Since these intersect at the right - angled vertex, the right triangle 's orthocenter -- the intersection of its three altitudes -- coincides with the right - angled vertex. Pythagorean theorem Main article : Pythagorean theorem The Pythagorean theorem states that : In any right triangle, the area of the square whose side is the hypotenuse ( the side opposite the right angle ) is equal to the sum of the areas of the squares whose sides are the two legs ( the two sides that meet at a right angle ). This can be stated in equation form as a 2 + b 2 = c 2 ( displaystyle displaystyle a ^ ( 2 ) + b ^ ( 2 ) = c ^ ( 2 ) ) where c is the length of the hypotenuse, and a and b are the lengths of the remaining two sides. Pythagorean triples are integer values of a, b, c satisfying this equation. Inradius and circumradius Illustration of the Pythagorean Theorem The radius of the incircle of a right triangle with legs a and b and hypotenuse c is r = a + b − c 2 = a b a + b + c. ( displaystyle r = ( \frac ( a + b-c ) ( 2 ) ) = ( \frac ( ab ) ( a + b + c ) ). ) The radius of the circumcircle is half the length of the hypotenuse, R = c 2. ( displaystyle R = ( \frac ( c ) ( 2 ) ). ) Thus the sum of the circumradius and the inradius is half the sum of the legs : R + r = a + b 2. ( displaystyle R + r = ( \frac ( a + b ) ( 2 ) ). ) One of the legs can be expressed in terms of the inradius and the other leg as a = 2 r ( b − r ) b − 2 r. ( displaystyle displaystyle a = ( \frac ( 2r ( b-r ) ) ( b - 2r ) ). ) Characterizations A triangle ABC with sides a ≤ b < c ( displaystyle a leq b < c ), semiperimeter s, area T, altitude h opposite the longest side, circumradius R, inradius r, exradii r, r, r ( tangent to a, b, c respectively ), and medians m, m, m is a right triangle if and only if any one of the statements in the following six categories is true. All of them are of course also properties of a right triangle, since characterizations are equivalences. Sides and semiperimeter a 2 + b 2 = c 2 ( Pythagorean theorem ) ( displaystyle displaystyle a ^ ( 2 ) + b ^ ( 2 ) = c ^ ( 2 ) quad ( ( \text ( Pythagorean theorem ) ) ) ) ( s − a ) ( s − b ) = s ( s − c ) ( displaystyle displaystyle ( s-a ) ( s-b ) = s ( s-c ) ) s = 2 R + r. ( displaystyle displaystyle s = 2R + r. ) a 2 + b 2 + c 2 = 8 R 2. ( displaystyle displaystyle a ^ ( 2 ) + b ^ ( 2 ) + c ^ ( 2 ) = 8R ^ ( 2 ). ) Angles A and B are complementary. cos ⁡ A cos ⁡ B cos ⁡ C = 0. ( displaystyle displaystyle cos ( A ) cos ( B ) cos ( C ) = 0. ) sin 2 ⁡ A + sin 2 ⁡ B + sin 2 ⁡ C = 2. ( displaystyle displaystyle sin ^ ( 2 ) ( A ) + sin ^ ( 2 ) ( B ) + sin ^ ( 2 ) ( C ) = 2. ) cos 2 ⁡ A + cos 2 ⁡ B + cos 2 ⁡ C = 1. ( displaystyle displaystyle cos ^ ( 2 ) ( A ) + cos ^ ( 2 ) ( B ) + cos ^ ( 2 ) ( C ) = 1. ) sin ⁡ 2 A = sin ⁡ 2 B = 2 sin ⁡ A sin ⁡ B. ( displaystyle displaystyle sin ( 2A ) = sin ( 2B ) = 2 sin ( A ) sin ( B ). ) Area T = a b 2 ( displaystyle displaystyle T = ( \frac ( ab ) ( 2 ) ) ) T = r a r b = r r c ( displaystyle displaystyle T = r_ ( a ) r_ ( b ) = rr_ ( c ) ) T = r ( 2 R + r ) ( displaystyle displaystyle T = r ( 2R + r ) ) T = P A ⋅ P B, ( displaystyle T = PA cdot PB, ) where P is the tangency point of the incircle at the longest side AB. Inradius and exradii r = s − c = ( a + b − c ) / 2 ( displaystyle displaystyle r = s-c = ( a + b-c ) / 2 ) r a = s − b = ( a − b + c ) / 2 ( displaystyle displaystyle r_ ( a ) = s-b = ( a-b + c ) / 2 ) r b = s − a = ( − a + b + c ) / 2 ( displaystyle displaystyle r_ ( b ) = s-a = ( - a + b + c ) / 2 ) r c = s = ( a + b + c ) / 2 ( displaystyle displaystyle r_ ( c ) = s = ( a + b + c ) / 2 ) r a + r b + r c + r = a + b + c ( displaystyle displaystyle r_ ( a ) + r_ ( b ) + r_ ( c ) + r = a + b + c ) r a 2 + r b 2 + r c 2 + r 2 = a 2 + b 2 + c 2 ( displaystyle displaystyle r_ ( a ) ^ ( 2 ) + r_ ( b ) ^ ( 2 ) + r_ ( c ) ^ ( 2 ) + r ^ ( 2 ) = a ^ ( 2 ) + b ^ ( 2 ) + c ^ ( 2 ) ) r = r a r b r c ( displaystyle displaystyle r = ( \frac ( r_ ( a ) r_ ( b ) ) ( r_ ( c ) ) ) ) Altitude and medians h = a b c ( displaystyle displaystyle h = ( \frac ( ab ) ( c ) ) ) m a 2 + m b 2 + m c 2 = 6 R 2. ( displaystyle displaystyle m_ ( a ) ^ ( 2 ) + m_ ( b ) ^ ( 2 ) + m_ ( c ) ^ ( 2 ) = 6R ^ ( 2 ). ) The length of one median is equal to the circumradius. The shortest altitude ( the one from the vertex with the biggest angle ) is the geometric mean of the line segments it divides the opposite ( longest ) side into. This is the right triangle altitude theorem. Circumcircle and incircle The triangle can be inscribed in a semicircle, with one side coinciding with the entirety of the diameter ( Thales ' theorem ). The circumcenter is the midpoint of the longest side. The longest side is a diameter of the circumcircle ( c = 2 R ). ( displaystyle displaystyle ( c = 2R ). ) The circumcircle is tangent to the nine - point circle. The orthocenter lies on the circumcircle. The distance between the incenter and the orthocenter is equal to 2 r ( displaystyle ( sqrt ( 2 ) ) r ). Trigonometric ratios Main article : Trigonometric functions -- Right - angled triangle definitions The trigonometric functions for acute angles can be defined as ratios of the sides of a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are similar. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled O, A and H respectively, then the trigonometric functions are sin ⁡ α = O H, cos ⁡ α = A H, tan ⁡ α = O A, sec ⁡ α = H A, cot ⁡ α = A O, csc ⁡ α = H O. ( displaystyle sin alpha = ( \frac ( O ) ( H ) ), , cos alpha = ( \frac ( A ) ( H ) ), , \tan alpha = ( \frac ( O ) ( A ) ), , sec alpha = ( \frac ( H ) ( A ) ), , cot alpha = ( \frac ( A ) ( O ) ), , csc alpha = ( \frac ( H ) ( O ) ). ) For the expression of hyperbolic functions as ratio of the sides of a right triangle, see the hyperbolic triangle of a hyperbolic sector. Special right triangles Main article : Special right triangles The values of the trigonometric functions can be evaluated exactly for certain angles using right triangles with special angles. These include the 30 - 60 - 90 triangle which can be used to evaluate the trigonometric functions for any multiple of π / 6, and the 45 - 45 - 90 triangle which can be used to evaluate the trigonometric functions for any multiple of π / 4. Kepler triangle Let H, G, and A be the harmonic mean, the geometric mean, and the arithmetic mean of two positive numbers a and b with a > b. If a right triangle has legs H and G and hypotenuse A, then A H = A 2 G 2 = G 2 H 2 = φ ( displaystyle ( \frac ( A ) ( H ) ) = ( \frac ( A ^ ( 2 ) ) ( G ^ ( 2 ) ) ) = ( \frac ( G ^ ( 2 ) ) ( H ^ ( 2 ) ) ) = phi , ) and a b = φ 3, ( displaystyle ( \frac ( a ) ( b ) ) = phi ^ ( 3 ), , ) where φ ( displaystyle phi ) is the golden ratio 1 + 5 2. ( displaystyle ( \tfrac ( 1 + ( sqrt ( 5 ) ) ) ( 2 ) ). , ) Since the sides of this right triangle are in geometric progression, this is the Kepler triangle. Thales ' theorem Main article : Thales ' theorem Median of a right angle of a triangle Thales ' theorem states that if A is any point of the circle with diameter BC ( except B or C themselves ) ABC is a right triangle where A is the right angle. The converse states that if a right triangle is inscribed in a circle then the hypotenuse will be a diameter of the circle. A corollary is that the length of the hypotenuse is twice the distance from the right angle vertex to the midpoint of the hypotenuse. Also, the center of the circle that circumscribes a right triangle is the midpoint of the hypotenuse and its radius is one half the length of the hypotenuse. Medians The following formulas hold for the medians of a right triangle : m a 2 + m b 2 = 5 m c 2 = 5 4 c 2. ( displaystyle m_ ( a ) ^ ( 2 ) + m_ ( b ) ^ ( 2 ) = 5m_ ( c ) ^ ( 2 ) = ( \frac ( 5 ) ( 4 ) ) c ^ ( 2 ). ) The median on the hypotenuse of a right triangle divides the triangle into two isosceles triangles, because the median equals one - half the hypotenuse. The medians m and m from the legs satisfy 4 c 4 + 9 a 2 b 2 = 16 m a 2 m b 2. ( displaystyle 4c ^ ( 4 ) + 9a ^ ( 2 ) b ^ ( 2 ) = 16m_ ( a ) ^ ( 2 ) m_ ( b ) ^ ( 2 ). ) Euler line In a right triangle, the Euler line contains the median on the hypotenuse -- that is, it goes through both the right - angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle 's orthocenter, the intersection of its altitudes, falls on the right - angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse. Inequalities In any right triangle the diameter of the incircle is less than half the hypotenuse, and more strongly it is less than or equal to the hypotenuse times ( 2 − 1 ). ( displaystyle ( ( sqrt ( 2 ) ) - 1 ). ) In a right triangle with legs a, b and hypotenuse c, c ≥ 2 2 ( a + b ) ( displaystyle c geq ( \frac ( sqrt ( 2 ) ) ( 2 ) ) ( a + b ) ) with equality only in the isosceles case. If the altitude from the hypotenuse is denoted h, then h c ≤ 2 4 ( a + b ) ( displaystyle h_ ( c ) leq ( \frac ( sqrt ( 2 ) ) ( 4 ) ) ( a + b ) ) with equality only in the isosceles case. Other properties If segments of lengths p and q emanating from vertex C trisect the hypotenuse into segments of length c / 3, then p 2 + q 2 = 5 ( c 3 ) 2. ( displaystyle p ^ ( 2 ) + q ^ ( 2 ) = 5 left ( ( \frac ( c ) ( 3 ) ) \right ) ^ ( 2 ). ) The right triangle is the only triangle having two, rather than one or three, distinct inscribed squares. Let h and k ( h > k ) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then 1 c 2 + 1 h 2 = 1 k 2. ( displaystyle ( \frac ( 1 ) ( c ^ ( 2 ) ) ) + ( \frac ( 1 ) ( h ^ ( 2 ) ) ) = ( \frac ( 1 ) ( k ^ ( 2 ) ) ). ) These sides and the incircle radius r are related by a similar formula : 1 r = − 1 c + 1 h + 1 k. ( displaystyle displaystyle ( \frac ( 1 ) ( r ) ) = - ( \frac ( 1 ) ( c ) ) + ( \frac ( 1 ) ( h ) ) + ( \frac ( 1 ) ( k ) ). ) The perimeter of a right triangle equals the sum of the radii of the incircle and the three excircles : a + b + c = r + r a + r b + r c. ( displaystyle a + b + c = r + r_ ( a ) + r_ ( b ) + r_ ( c ). ) See also Acute and obtuse triangles ( oblique triangles ) References Jump up ^ Di Domenico, Angelo S., `` A property of triangles involving area '', Mathematical Gazette 87, July 2003, pp. 323 - 324. ^ Jump up to : Posamentier, Alfred S., and Salkind, Charles T. Challenging Problems in Geometry, Dover, 1996. Jump up ^ Wentworth p. 156 Jump up ^ Voles, Roger, `` Integer solutions of a − 2 + b − 2 = d − 2 ( displaystyle a ^ ( - 2 ) + b ^ ( - 2 ) = d ^ ( - 2 ) ), '' Mathematical Gazette 83, July 1999, 269 -- 271. Jump up ^ Richinick, Jennifer, `` The upside - down Pythagorean Theorem, '' Mathematical Gazette 92, July 2008, 313 -- 317. ^ Jump up to : Inequalities proposed in `` Crux Mathematicorum '', ( 1 ). Jump up ^ Triangle right iff s = 2R + r, Art of problem solving, 2011 ^ Jump up to : Andreescu, Titu and Andrica, Dorian, `` Complex Numbers from A to... Z '', Birkhäuser, 2006, pp. 109 - 110. Jump up ^ Properties of Right Triangles ^ Jump up to : CTK Wiki Math, A Variant of the Pythagorean Theorem, 2011, ( 2 ). Jump up ^ Darvasi, Gyula ( March 2005 ), `` Converse of a Property of Right Triangles '', The Mathematical Gazette, 89 ( 514 ) : 72 -- 76. Jump up ^ Bell, Amy ( 2006 ), `` Hansen 's Right Triangle Theorem, Its Converse and a Generalization '' ( PDF ), Forum Geometricorum, 6 : 335 -- 342 Jump up ^ Di Domenico, A., `` The golden ratio -- the right triangle -- and the arithmetic, geometric, and harmonic means, '' Mathematical Gazette 89, July 2005, 261. Also Mitchell, Douglas W., `` Feedback on 89.41 '', vol 90, March 2006, 153 - 154. ^ Jump up to : Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles. Prometheus Books, 2012. Jump up ^ Bailey, Herbert, and DeTemple, Duane, `` Squares inscribed in angles and triangles '', Mathematics Magazine 71 ( 4 ), 1998, 278 - 284. Weisstein, Eric W. `` Right Triangle ''. MathWorld. Wentworth, G.A. ( 1895 ). A Text - Book of Geometry. Ginn & Co. External links Wikimedia Commons has media related to Right triangles. 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Question: what is the opposite side of a right angle triangle
Answer: hypotenuse