What is how to find the hypotenuse of a triangle?

The hypotenuse is the longest side of a <a href="https://www.wikiwhat.page/kavramlar/right%20triangle">right triangle</a>, and it is opposite the right angle. There are a few ways to find the length of the hypotenuse:

• Using the Pythagorean Theorem: This is the most common method. 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 other two sides (a and b). The formula is: a² + b² = c². To find the hypotenuse, you need to:

  1. Square the lengths of the two shorter sides (legs) of the right triangle (a and b).
  2. Add the squares together.
  3. Take the square root of the sum. This gives you the length of the hypotenuse (c).

• Using Trigonometry: If you know one of the acute angles (other than the right angle) and the length of one of the legs, you can use trigonometric functions like <a href="https://www.wikiwhat.page/kavramlar/sine">sine</a>, <a href="https://www.wikiwhat.page/kavramlar/cosine">cosine</a>, or <a href="https://www.wikiwhat.page/kavramlar/tangent">tangent</a> to find the hypotenuse.

  • If you know an angle (θ) and the length of the side opposite to it (a), use: sin(θ) = a / c, so c = a / sin(θ).
  • If you know an angle (θ) and the length of the side adjacent to it (b), use: cos(θ) = b / c, so c = b / cos(θ).

• Special Right Triangles: Certain right triangles have special angle and side relationships that make finding the hypotenuse easier:

  • 45-45-90 Triangle: In this type of triangle, the two legs are equal in length (let's call it 'a'). The hypotenuse is then equal to a√2.
  • 30-60-90 Triangle: In this type of triangle, if the side opposite the 30-degree angle is 'a', then the hypotenuse is 2a, and the side opposite the 60-degree angle is a√3.