Pythagorean Theorem Basics
The Pythagorean Theorem relates the sides of a right triangle, a cornerstone of geometry discovered by the ancient Greek mathematician Pythagoras (though used earlier by others too).
The Formula
For a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\):
\[ a^2 + b^2 = c^2 \]
Example: 3-4-5 Triangle
If \(a = 3\), \(b = 4\):
\[ 3^2 + 4^2 = 9 + 16 = 25 \]
\[ c^2 = 25 \Rightarrow c = \sqrt{25} = 5 \]
Verification: A triangle with sides 3, 4, and 5 is indeed a right triangle.
Proof by Area
Imagine a square with side length \(a + b\). Place four right triangles (legs \(a\) and \(b\)) inside it, leaving a smaller square with side \(c\). The total area is \( (a + b)^2 \), and the area of the triangles plus the inner square is:
\[ (a + b)^2 = 4 \cdot \frac{1}{2}ab + c^2 \]
\[ a^2 + 2ab + b^2 = 2ab + c^2 \]
\[ a^2 + b^2 = c^2 \]
Applications
Use it to find distances (e.g., diagonal of a room), in navigation, or to verify right angles in construction.