The Pythagorean Theorem is a fundamental principle in geometry, describing the relationship between the sides of a right triangle. It states:

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

Explanation:

• $a$ and $b$ are the lengths of the two legs (shorter sides) of the right triangle.
• $c$ is the length of the hypotenuse (the side opposite the right angle).

The theorem is used to:

1. Calculate the length of one side of a right triangle if the other two sides are known.
2. Prove that a triangle is a right triangle by checking the equality of the sides.

Example:

If a right triangle has legs $a = 3$ and $b = 4$, the hypotenuse $c$ can be calculated as:

$c^2 = a^2 + b^2$ $c^2 = 3^2 + 4^2 = 9 + 16 = 25$ $c = \sqrt{25} = 5$

Thus, the hypotenuse is $5$.

Would you like a detailed derivation, a proof, or example problems solved?

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Questions to Explore:

1. How can the Pythagorean Theorem be applied in three dimensions?
2. What is the converse of the Pythagorean Theorem, and how is it used?
3. Can the theorem be extended to non-Euclidean geometries?
4. How do you derive the Pythagorean Theorem using coordinate geometry?
5. What are the real-world applications of the Pythagorean Theorem?

Tip:

When working with the Pythagorean Theorem, always confirm the triangle is a right triangle before applying the