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!

Learn how to use the Pythagorean theorem to solve for missing sides in right-angled triangles. Includes step-by-step instructions, formulas, and examples, making it ideal for students in Grades 7-9.

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