Which of the following sets of numbers could not represent the three sides of a right triangle? Answer \( \begin{array}{ll}\{9,12,15\} & \{4,12,13\} \\ \{45,60,75\} & \{12,35,37\}\end{array} \)

To determine which set of numbers cannot represent the sides of a right triangle, we can use the Pythagorean theorem. For a set \(\{a, b, c\}\) where \(c\) is the largest number, the set forms a right triangle if and only if: \[ a^2 + b^2 = c^2 \] Let's evaluate each set: 1. **\(\{9, 12, 15\}\)** \[ 9^2 + 12^2 = 81 + 144 = 225 = 15^2 \] This set satisfies the Pythagorean theorem and **can** represent a right triangle. 2. **\(\{4, 12, 13\}\)** \[ 4^2 + 12^2 = 16 + 144 = 160 \neq 13^2 (169) \] This set does **not** satisfy the Pythagorean theorem and **cannot** represent a right triangle. 3. **\(\{45, 60, 75\}\)** \[ 45^2 + 60^2 = 2025 + 3600 = 5625 = 75^2 \] This set satisfies the Pythagorean theorem and **can** represent a right triangle. 4. **\(\{12, 35, 37\}\)** \[ 12^2 + 35^2 = 144 + 1225 = 1369 = 37^2 \] This set satisfies the Pythagorean theorem and **can** represent a right triangle. **Conclusion:** The set \(\{4, 12, 13\}\) does **not** satisfy the Pythagorean theorem and therefore cannot represent the sides of a right triangle. **Answer:** \(\{4,12,13\}\)