The **Pythagorean Theorem** is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states:
> **In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs).**
### Formula:
\[
\boxed{a^2 + b^2 = c^2}
\]
Where:
- \( c \) = hypotenuse (longest side)
- \( a \) and \( b \) = other two sides (legs)
### Example:
If a right triangle has legs of lengths \( 3 \) and \( 4 \), the hypotenuse \( c \) is calculated as:
\[
3^2 + 4^2 = c^2 \\
9 + 16 = c^2 \\
25 = c^2 \\
c = \sqrt{25} = 5
\]
This theorem is named after the ancient Greek mathematician **Pythagoras**, though it was known to Babylonians and Indians earlier. It has wide applications in mathematics, physics, engineering, and more.
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Welcome to our exploration of the Pythagorean Theorem! This fundamental principle in geometry describes the relationship between the sides of a right triangle. As we can see here, we have a right triangle with sides a, b, and the hypotenuse c. The theorem states that the square of the hypotenuse equals the sum of squares of the other two sides.