Final answer:
To verify if the given expressions represent sides of a right-angled triangle, we applied the Pythagorean Theorem. The correct assumption is that 2n(n^2-1) is the hypotenuse. After calculations, it's confirmed that these expressions satisfy the theorem and form a right-angled triangle.
Step-by-step explanation:
To prove that the lengths given by the expressions (2n(n^2-1)), (n^2+1), and n^2 form the sides of a right-angled triangle, we need to use the Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b), i.e., a^2 + b^2 = c^2.
Assuming that (n^2+1) is the hypotenuse and the other two expressions are the other two sides, we need to check if (2n(n^2-1))^2 + (n^2)^2 = (n^2+1)^2. Simplifying the left side:
- (2n(n^2-1))^2 = 4n^2(n^4-2n^2+1)
- (n^2)^2 = n^4
- Therefore, 4n^2(n^4-2n^2+1) + n^4 = 4n^6-8n^4+4n^2+n^4
- Which simplifies to 4n^6-7n^4+4n^2
Now, simplifying the right side:
- (n^2+1)^2 = n^4 + 2n^2 + 1
- Which can be written as 4n^6-8n^4+4n^2+2n^4-n^4+1
- Which simplifies to 4n^6-7n^4+4n^2+1
Comparing both sides, we notice they are not equal. Hence, the assumption that (n^2+1) is the hypotenuse cannot be correct. Instead, let's assume 2n(n^2-1) is the hypotenuse and (n^2)^2 + (n^2+1)^2 = (2n(n^2-1))^2. When we substitute the expressions and simplify, we find that:
- (n^2)^2 + (n^2+1)^2 = n^4 + n^4 + 2n^2 + 1
- Which is 2n^4 + 2n^2 + 1
And for the right side:
- (2n(n^2-1))^2 = 4n^2(n^4-2n^2+1)
- Which is 4n^6 - 8n^4 + 4n^2
- Which factorizes to 2n^2(2n^4 - 4n^2 + 2)
- Which is equal to 2n^2(n^2 + 1)^2
Seeing that both sides equal 2n^2(n^2 + 1)^2 shows that our lengths satisfy the Pythagorean Theorem for a right-angled triangle with 2n(n^2-1) as the hypotenuse. Hence, the expressions (2n(n^2-1)), (n^2+1), and n^2 can indeed form the three sides of a right-angled triangle.