To determine which set of numbers does not represent the sides of a right-angled triangle, we rely on a fundamental principle in geometry known as the Pythagorean theorem. This theorem applies specifically to right-angled triangles and states that the square of the length of the longest side (called the hypotenuse) is equal to the sum of the squares of the lengths of the other two shorter sides (called legs).

In mathematical terms, if 'a' and 'b' are the lengths of the two legs and 'c' is the length of the hypotenuse, then the relationship is:

$$\text{a}^2 + \text{b}^2 = \text{c}^2$$

We will examine each given set of numbers. For each set, we must first identify the longest side, which will serve as 'c'. Then, we will check if the sum of the squares of the other two sides equals the square of the longest side.

Triangle Side Set Analysis

Let's analyze each option provided:

Right-angled Triangle Check: Option 1 (12, 16, 20)

• The lengths of the sides are 12, 16, and 20.
• The longest side is 20, so we set $\text{c} = 20$.
• The other two sides are $\text{a} = 12$ and $\text{b} = 16$.
• Now, we apply the Pythagorean theorem:
• $$\text{a}^2 + \text{b}^2 = 12^2 + 16^2 = 144 + 256 = 400$$
• $$\text{c}^2 = 20^2 = 400$$
• Since $400 = 400$, this set of numbers does represent the sides of a right-angled triangle.

Right-angled Triangle Check: Option 2 (9, 5, 6)

• The lengths of the sides are 9, 5, and 6.
• The longest side is 9, so we set $\text{c} = 9$.
• The other two sides are $\text{a} = 5$ and $\text{b} = 6$.
• Now, we apply the Pythagorean theorem:
• $$\text{a}^2 + \text{b}^2 = 5^2 + 6^2 = 25 + 36 = 61$$
• $$\text{c}^2 = 9^2 = 81$$
• Since $61 \ne 81$, this set of numbers does not represent the sides of a right-angled triangle.

Right-angled Triangle Check: Option 3 (6, 10, 8)

• The lengths of the sides are 6, 10, and 8.
• The longest side is 10, so we set $\text{c} = 10$.
• The other two sides are $\text{a} = 6$ and $\text{b} = 8$.
• Now, we apply the Pythagorean theorem:
• $$\text{a}^2 + \text{b}^2 = 6^2 + 8^2 = 36 + 64 = 100$$
• $$\text{c}^2 = 10^2 = 100$$
• Since $100 = 100$, this set of numbers does represent the sides of a right-angled triangle.

Right-angled Triangle Check: Option 4 (15, 25, 20)

• The lengths of the sides are 15, 25, and 20.
• The longest side is 25, so we set $\text{c} = 25$.
• The other two sides are $\text{a} = 15$ and $\text{b} = 20$.
• Now, we apply the Pythagorean theorem:
• $$\text{a}^2 + \text{b}^2 = 15^2 + 20^2 = 225 + 400 = 625$$
• $$\text{c}^2 = 25^2 = 625$$
• Since $625 = 625$, this set of numbers does represent the sides of a right-angled triangle.

Triangle Sides Summary

Here is a summary of our findings for each set of sides:

Conclusion

Based on our application of the Pythagorean theorem, the set of sides 9, 5, 6 is the only combination that does not satisfy the condition $\text{a}^2 + \text{b}^2 = \text{c}^2$. Therefore, this set does not represent the sides of a right-angled triangle.