Definition:Triangle (Geometry)/Right-Angled

Definition

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

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.

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 $21$)

Also known as

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

Also see

• Pythagoras's Theorem

• Results about right triangles can be found here.

Sources

• 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): triangle (Euclidean geometry)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): right-angled triangle
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): triangle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): right-angled triangle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): triangle
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): right-angled triangle