Definition:Triangle (Geometry)/Right-Angled

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Definition

A right-angled triangle is a triangle in which one of the vertices is a right angle.

RightTriangle.png

Note that in order to emphasise the nature of the right angle in such a triangle, a small square is usually drawn inside it.


Note the following nomenclature in the context of a right-angled triangle:


Hypotenuse

In a right-angled triangle, the opposite side to the right angle is known as the hypotenuse.

In the above figure, the side labeled $b$ is the hypotenuse.


Adjacent

In a right-angled triangle, for a given non-right angled vertex, the adjacent side which is not the hypotenuse is referred to as the adjacent.

In the above figure:

the adjacent to vertex $A$ is side $c$
the adjacent to vertex $C$ is side $a$.


Opposite

In a right-angled triangle, for a given non-right angled vertex, the opposite side is referred to as the opposite.

In the above figure:

the opposite to vertex $A$ is side $a$
the opposite to vertex $C$ is side $c$.


Legs

In a right-angled triangle, the two sides which are not the hypotenuse are referred to as its legs.


Also known as

A right-angled triangle is also often referred to as a right triangle.


Euclid's Definition

In the words of Euclid:

Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

(The Elements: Book $\text{I}$: Definition $21$)