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.
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
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.
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
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
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.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
This can be stated in equation form as
where c is the length of the hypotenuse, and a and b are the lengths of the remaining two sides.
The radius of the incircle of a right triangle with legs a and b and hypotenuse c is
r=a+b−c2=aba+b+c.{\displaystyle r={\frac {a+b-c}{2}}={\frac {ab}{a+b+c}}.}circumcircle is half the length of the hypotenuse,
R=c2.{\displaystyle R={\frac {c}{2}}.}
Thus the sum of the circumradius and the inradius is half the sum of the legs:[6]
R+r=Thus the sum of the circumradius and the inradius is half the sum of the legs:[6]
R+r=a+b2.{\displaystyle R+r={\frac {a+b}{2}}.}a=2r(b−r)b−2r.A triangle ABC with sides a≤b<c{\displaystyle a\leq b<c}, semiperimeters, areaT, altitudeh opposite the longest side, circumradiusR, inradiusr, exradiira, rb, rc (tangent to a, b, c respectively), and mediansma, mb, mc 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
a2+b2=c2(Pythagorean theorem){\displaystyle \displaystyle a^{2}+b^{2}=c^{2}\quad ({\text{Pythagorean theorem}})}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
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[13]
where ϕ{\displaystyle \phi } is the golden ratio1+52.{\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}.\,} Since the sides of this right triangle are in geometric progression, this is the Kepler 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:
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).}[14]:p.281
In a right triangle with legs a, b and hypotenuse c,
^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.