Solving Right Triangles Using the Pythagorean Theorem

Right Triangles and the Pythagorean Theorem

Solving for the Hypotenuse

The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that 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).

• Pythagorean Theorem:

$c^2 = a^2 + b^2$

• Solving for the Hypotenuse: To find the length of the hypotenuse, take the square root of the sum of the squares of the other two sides.

$c = \sqrt{a^2 + b^2}$

• Example: If a right triangle has legs of length 44 and 117, the hypotenuse is found as follows:

$c = \sqrt{44^2 + 117^2}$ $c = \sqrt{1936 + 13689}$ $c = \sqrt{15625}$ $c = 125$

• Application: This method is used to solve for the missing side in any right triangle when two sides are known.

Key Points

• The Pythagorean Theorem applies only to right triangles.

• Always check that the triangle is right-angled before applying the theorem.

• The hypotenuse is always the side opposite the right angle and is the longest side of the triangle.

Additional info:

• This topic is foundational for later trigonometric applications, such as defining sine, cosine, and tangent in terms of triangle sides.