The question asks which set of side lengths can be used to prove that the statement “In any given triangle, the square of the longest side is equal to the sum of the squares of the two shorter sides” is false. This statement is the Pythagorean Theorem, which is true for right triangles but false for other types of triangles (acute or obtuse).

We need to check each set of side lengths to see if the Pythagorean Theorem holds or fails:

1. Option A: 3 cm, 4 cm, 5 cm $5^2 = 3^2 + 4^2 \implies 25 = 9 + 16 \implies 25 = 25 \quad (\text{True, it's a right triangle})$

2. Option B: 4 cm, 5 cm, 7 cm $7^2 = 4^2 + 5^2 \implies 49 \neq 16 + 25 \implies 49 \neq 41 \quad (\text{False, not a right triangle})$

3. Option C: 5 cm, 12 cm, 13 cm $13^2 = 5^2 + 12^2 \implies 169 = 25 + 144 \implies 169 = 169 \quad (\text{True, it's a right triangle})$

4. Option D: 9 cm, 12 cm, 15 cm $15^2 = 9^2 + 12^2 \implies 225 = 81 + 144 \implies 225 = 225 \quad (\text{True, it's a right triangle})$

Thus, Option B (4 cm, 5 cm, 7 cm) proves the statement false, since the Pythagorean Theorem does not hold for this triangle.

Would you like further details or have any questions?

Here are some related questions:

1. What is the Pythagorean Theorem and when is it applied?
2. Can the Pythagorean Theorem be applied to non-right triangles?
3. How can you identify an obtuse triangle using side lengths?
4. What is the general formula for checking if a triangle is acute, obtuse, or right-angled using sides?
5. How do you prove a triangle is a right triangle using side lengths?

Tip: If the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is obtuse!