Properties of a Right Triangle
A right triangle is a type of triangle that has one angle that measures exactly 90°. Right triangles, and the relationships between their sides and angles, form the basis of trigonometry.
In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides (legs).
Pythagorean Theorem
The most famous relationship in a right triangle is the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides: a² + b² = c². If all three sides have lengths that are integers, they are collectively known as a Pythagorean triple. Common examples include 3-4-5, 5-12-13, and 8-15-17.
Variables
- a, bLegs (shorter sides)
- cHypotenuse (longest side)
- α, βNon-right angles (α + β = 90°)
- hAltitude to the hypotenuse
Area, Perimeter, and Altitude
Area and perimeter of a right triangle are calculated in the same way as any other triangle. The perimeter is the sum of the three sides. The area can be determined using the formula: Area = ½ × a × b or Area = ½ × c × h.
The variable h refers to the altitude of the triangle, which is the length from the vertex of the right angle to the hypotenuse. The altitude divides the original triangle into two smaller triangles that are mathematically similar to the original triangle.
Special Right Triangles
Certain right triangles have specific properties that make calculations much easier without evaluating trigonometric functions:
30°-60°-90° Triangle
The sides corresponding to the angles 30°-60°-90° follow a ratio of 1 : √3 : 2. If the length of one side is known, the other sides can be determined using this ratio immediately. For instance, the hypotenuse is exactly twice the length of the shortest side.
45°-45°-90° Triangle
Also referred to as an isosceles right triangle since it has two sides of equal lengths. Its sides follow a ratio of 1 : 1 : √2. The hypotenuse can be found simply by multiplying the leg length by the square root of 2.