Which set of side lengths can be used to prove the statement below is false?

“In any given triangle, the square of the longest side is equal to the sum of the squares of the two shorter sides.”

A.3 cm, 4 cm, 5 cm

B.4 cm, 5 cm, 7 cm

C.5 cm, 12 cm, 13 cm

D.9 cm, 12 cm, 15 cm

The statement in question is actually the **Pythagorean Theorem**, which states that in a **right triangle**, the square of the longest side (the hypotenuse) is equal to the sum of the squares of the two shorter sides. This theorem only holds true for right triangles, not for all triangles. To disprove the statement, we need to find a set of side lengths that **do not satisfy the Pythagorean Theorem**.

Let's analyze each option using the Pythagorean Theorem:  
For a triangle with side lengths \(a\), \(b\), and \(c\) where \(c\) is the longest side:
\[
a^2 + b^2 = c^2
\]

### A. \(3\), \(4\), \(5\)
- \(3^2 + 4^2 = 9 + 16 = 25\)
- \(5^2 = 25\)
- **True**: This satisfies the Pythagorean Theorem.

### B. \(4\), \(5\), \(7\)
- \(4^2 + 5^2 = 16 + 25 = 41\)
- \(7^2 = 49\)
- **False**: This does not satisfy the Pythagorean Theorem, so this set of sides **disproves the statement**.

### C. \(5\), \(12\), \(13\)
- \(5^2 + 12^2 = 25 + 144 = 169\)
- \(13^2 = 169\)
- **True**: This satisfies the Pythagorean Theorem.

### D. \(9\), \(12\), \(15\)
- \(9^2 + 12^2 = 81 + 144 = 225\)
- \(15^2 = 225\)
- **True**: This satisfies the Pythagorean Theorem.

### Conclusion:
The set of side lengths **4 cm, 5 cm, 7 cm** (option **B**) proves the statement false, as it does not satisfy the Pythagorean Theorem.

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Would you like more details or have any questions?

Here are 5 related questions:
1. What is the geometric significance of the Pythagorean Theorem?
2. Can the Pythagorean Theorem be used for non-right triangles?
3. How do you determine whether a triangle is right, obtuse, or acute using its sides?
4. What is the converse of the Pythagorean Theorem?
5. What is the importance of the Pythagorean triplets in geometry?

**Tip**: The Pythagorean Theorem only applies to right triangles, so always ensure the triangle is right-angled before applying it!

Explore how to use side lengths to disprove the Pythagorean Theorem. This problem demonstrates how to apply the theorem to analyze sets of side lengths and identify if a triangle is right-angled or not. Suitable for Grades 8-10.

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