The Pythagorean Theorem is a fundamental principle in geometry that applies to right-angled triangles. It relates the lengths of the three sides of the triangle:

• Hypotenuse (c): the side opposite the right angle (the longest side).
• Legs (a and b): the two sides that form the right angle.

The Pythagorean Theorem states that:

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

Where:

• $a$ and $b$ are the lengths of the two legs,
• $c$ is the length of the hypotenuse.

How to use the Pythagorean Theorem:

1. Given the lengths of two sides, you can find the third side:
   • If you have the two legs (a and b), you can find the hypotenuse (c) by rearranging the formula: $c = \sqrt{a^2 + b^2}$
   • If you have one leg and the hypotenuse, you can find the other leg by rearranging the formula: $a = \sqrt{c^2 - b^2} \quad \text{or} \quad b = \sqrt{c^2 - a^2}$

Example:

If you have a right triangle with legs of length 3 and 4, and you need to find the hypotenuse: $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Thus, the hypotenuse is 5 units long.

Would you like an example with numbers to solve, or do you have any questions about this theorem?

Here are 5 related questions:

1. How can the Pythagorean theorem be applied in real-life situations?
2. What happens if you use the Pythagorean theorem with non-right-angled triangles?
3. How can the theorem be extended to 3D geometry?
4. Can the theorem be used to find distances in coordinate geometry?
5. How do you check if a triangle is a right triangle using the Pythagorean theorem?

Tip: Always double-check if the triangle is right-angled before applying the Pythagorean theorem!