Right triangle

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Illustration to Euclid's proof of the Pythagorean theorem2
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Right triangle or right-angled triangle is a triangle in which one of the angles is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry.

The side opposite the right angle is called the hypotenuse (side c in the figure below). The sides adjacent to the right angle are called the legs (sides a and b in the figure below).

Properties and formulas[edit | edit source]

A right triangle has several important properties and formulas:

  • 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): a² + b² = c².
  • The trigonometric functions (sine, cosine, and tangent) can be defined in the context of a right triangle. For an angle θ:
    • The sine of θ is the ratio of the length of the opposite leg to the length of the hypotenuse.
    • The cosine of θ is the ratio of the length of the adjacent leg to the length of the hypotenuse.
    • The tangent of θ is the ratio of the length of the opposite leg to the length of the adjacent leg.
  • The area of a right triangle can be found using the formula: 1/2 * base * height, where the base and height are the lengths of the two legs.

Special right triangles[edit | edit source]

There are two special types of right triangles:

  • A 45°-45°-90° triangle, where the legs are of equal length and the hypotenuse is √2 times the length of a leg.
  • A 30°-60°-90° triangle, where the lengths of the sides are in the ratio 1:√3:2, respectively.

Applications[edit | edit source]

Right triangles are used in various fields such as engineering, architecture, and physics, where the principles of trigonometry are applied to solve problems involving distances, angles, and heights.

See also[edit | edit source]

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Contributors: Prab R. Tumpati, MD