Right triangle

A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a <a href="/facts/Triangle/cRXUwsMn">triangle</a> in which two <a href="/facts/Edge_(geometry)/3lFN6sEz">sides</a> are <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a>, forming a <a href="/facts/Right_angle/psaslRq7">right angle</a> (1⁄4 <a href="/facts/Turn_(unit)/QczxAyuM">turn</a> or 90 <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a>).
The side opposite to the right angle is called the <a href="/facts/Hypotenuse/hVMauM5f">hypotenuse</a> (side 
 
 
 
 c
 
 
 {\displaystyle c}
 
 in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: <a href="/facts/Cathetus/jApv1bHv">cathetus</a>). Side 
 
 
 
 a
 
 
 {\displaystyle a}
 
 may be identified as the side adjacent to angle 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and opposite (or opposed to) angle 
 
 
 
 A
 ,
 
 
 {\displaystyle A,}
 
 while side 
 
 
 
 b
 
 
 {\displaystyle b}
 
 is the side adjacent to angle 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and opposite angle 
 
 
 
 B
 .
 
 
 {\displaystyle B.}

Every right triangle is half of a <a href="/facts/Rectangle/mdLfIGKA">rectangle</a> which has been divided along its <a href="/facts/Diagonal/39z9sIRb">diagonal</a>. When the rectangle is a <a href="/facts/Square/JHJeMvmL">square</a>, its right-triangular half is <a href="/facts/Isosceles_triangle/rvLe3yqK">isosceles</a>, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is <a href="/facts/Scalene_triangle/cRXUwsMn">scalene</a>.
Every triangle whose <a href="/facts/Base_(geometry)/YkaFWKCF">base</a> is the <a href="/facts/Diameter/6uNn4VmU">diameter</a> of a <a href="/facts/Circle/9fy5ggKn">circle</a> and whose <a href="/facts/Apex_(geometry)/SrMxmHJk">apex</a> lies on the circle is a right triangle, with the right angle at the apex and the hypotenuse as the base; conversely, the <a href="/facts/Circumcircle/RafYT4am">circumcircle</a> of any right triangle has the hypotenuse as its diameter. This is <a href="/facts/Thales%2527_theorem/wI6hpFFk">Thales' theorem</a>.
The legs and hypotenuse of a right triangle satisfy the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a>: the sum of the areas of the squares on two legs is the area of the square on the hypotenuse, 
 
 
 
 
 a
 
 2
 
 
 +
 
 b
 
 2
 
 
 =
 
 c
 
 2
 
 
 .
 
 
 {\displaystyle a^{2}+b^{2}=c^{2}.}
 
 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 <a href="/facts/Pythagorean_triple/I3IJgWhf">Pythagorean triple</a>.
The relations between the sides and angles of a right triangle provides one way of defining and understanding <a href="/facts/Trigonometry/K4hfaC6F">trigonometry</a>, the study of the metrical relationships between lengths and angles.

Right triangle open-in-new

Right triangle