Definition and Formula
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides (the legs) is equal to the square of the longest side (the hypotenuse).
Mathematically, the formula is expressed as:
\(a^2 + b^2 = c^2\)
- \(a\) and \(b\) are the lengths of the legs.
- \(c\) is the length of the hypotenuse.
The theorem is named after Pythagoras, an ancient Greek mathematician. This principle is fundamental in various disciplines, including engineering, physics, and architecture.
Applications of the Pythagorean Theorem
- Calculating distances between two points on a coordinate plane.
- Determining heights of structures like buildings or trees.
- Designing stable structures with precise right angles.
Examples
Example 1: Finding the Hypotenuse
Given a right triangle with legs \(a = 3\) and \(b = 4\), calculate the hypotenuse:
\(c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = 5\)
Example 2: Height of a Ladder
A ladder leans against a wall. Its base is 6 feet from the wall, and the ladder is 8 feet long. Find the height where the ladder touches the wall:
\(b = \sqrt{c^2 - a^2} = \sqrt{8^2 - 6^2} = \sqrt{28}\)
Example 3: Diagonal of a Square
A square has a diagonal of 10 units. Find the length of its sides:
\(a^2 + a^2 = 10^2 \Rightarrow a^2 = 50 \Rightarrow a = \sqrt{50} = 5\sqrt{2}\)
Example 4: Area to Diagonal
A square has an area of 25 square units. Find the length of its diagonal:
\(a^2 = 25 \Rightarrow a = 5\)
\(d^2 = 2a^2 = 2(5^2) = 50 \Rightarrow d = 5\sqrt{2}\)