Questions: Which of the following sets of numbers could not represent the three sides of a right triangle?

Solution Steps

To determine which set of numbers could not represent the sides of a right triangle, we can use the converse of the Pythagorean Theorem. For a set of three numbers \(a\), \(b\), and \(c\) (where \(c\) is the largest), they form a right triangle if and only if \(a^2 + b^2 = c^2\). We will check this condition for each set of numbers.

We are given the following sets of numbers to evaluate whether they can represent the sides of a right triangle:

1. \( \{57, 76, 95\} \)
2. \( \{18, 80, 82\} \)
3. \( \{10, 25, 26\} \)
4. \( \{65, 72, 97\} \)

For each set, we will check if the largest number squared equals the sum of the squares of the other two numbers. Specifically, for a set \( \{a, b, c\} \) where \( c \) is the largest, we need to verify if: \[ a^2 + b^2 = c^2 \]

1. For \( \{57, 76, 95\} \): \[ 57^2 + 76^2 = 3249 + 5776 = 9025 \quad \text{and} \quad 95^2 = 9025 \quad \Rightarrow \quad \text{Valid} \]

2. For \( \{18, 80, 82\} \): \[ 18^2 + 80^2 = 324 + 6400 = 6724 \quad \text{and} \quad 82^2 = 6724 \quad \Rightarrow \quad \text{Valid} \]

3. For \( \{10, 25, 26\} \): \[ 10^2 + 25^2 = 100 + 625 = 725 \quad \text{and} \quad 26^2 = 676 \quad \Rightarrow \quad \text{Invalid} \]

4. For \( \{65, 72, 97\} \): \[ 65^2 + 72^2 = 4225 + 5184 = 9409 \quad \text{and} \quad 97^2 = 9409 \quad \Rightarrow \quad \text{Valid} \]

The set \( \{10, 25, 26\} \) does not satisfy the condition of the converse of the Pythagorean theorem and therefore could not represent the sides of a right triangle.

Final Answer