Right Triangles

A right triangle is a type of triangle that has one angle measuring 90 degrees (the right angle). The side opposite the right angle is the longest side, called the hypotenuse, and the other two sides are called the legs. Right triangles have special properties and are the basis for trigonometry.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the legs (a and b). This is often written as:

a² + b² = c²

This theorem provides a method for calculating the length of one side of a right triangle when the lengths of the other two sides are known.

Trigonometric Ratios

In a right triangle, the ratios of the lengths of two sides are related to the measures of the angles. These ratios are the basis for the trigonometric functions:

  • Sine (sin) of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • Cosine (cos) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
  • Tangent (tan) of an angle is the ratio of the sine of the angle to the cosine of the angle, which is equivalent to the ratio of the length of the opposite side to the length of the adjacent side.

Special Right Triangles

There are two types of right triangles that have additional special properties:

  1. 45-45-90 Triangle: This is a right triangle in which the two non-right angles each measure 45 degrees. The sides are in the ratio of 1:1:√2.

  2. 30-60-90 Triangle: This is a right triangle in which the non-right angles measure 30 and 60 degrees. The sides are in the ratio of 1:√3:2.

Applications

Right triangles have many applications in different fields, including architecture, astronomy, physics, engineering, and computer graphics. They are also used in trigonometry to define the trigonometric functions and to solve problems involving angles and distances.