The Pythagorean theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is expressed as:

\[ c^2 = a^2 + b^2 \]

where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.

### Example Problem
Suppose we have a right-angled triangle with sides \( a = 3 \) and \( b = 4 \). To find the hypotenuse \( c \), we apply the Pythagorean theorem:

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

So, the length of the hypotenuse \( c \) is 5.

### Steps to Solve a Problem Using the Pythagorean Theorem:
1. Identify the lengths of the two legs (\( a \) and \( b \)).
2. Square both of these lengths.
3. Add the squared lengths together.
4. Take the square root of the sum to find the length of the hypotenuse (\( c \)).

Would you like more details on this process or have any specific questions about using the Pythagorean theorem?

### Relative Questions
1. How do you apply the Pythagorean theorem in three-dimensional geometry?
2. What are the historical origins of the Pythagorean theorem?
3. How can the Pythagorean theorem be used in real-world applications?
4. What are Pythagorean triples and how are they related to the theorem?
5. How can the Pythagorean theorem be proved using algebraic methods?
6. Can the Pythagorean theorem be extended to non-Euclidean geometry?
7. What are some common mistakes students make when using the Pythagorean theorem?
8. How does the Pythagorean theorem relate to the distance formula in coordinate geometry?

### Tip
Always check that the triangle is right-angled before applying the Pythagorean theorem, as it only holds true for right-angled triangles.

Learn how to solve a Pythagorean theorem problem with a step-by-step solution. This problem demonstrates the application of the Pythagorean theorem in geometry, where you find the length of the hypotenuse in a right-angled triangle. Suitable for students in Grades 7-9.

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