1. **State the problem:** Given two sides of a triangle, find which of the given options can be the length of the third side. 2. **Formula and rule:** For any triangle with sides $a$, $b$, and $c$, the triangle inequality states: $$|a - b| < c < a + b$$ This means the third side must be greater than the absolute difference of the two given sides and less than their sum. --- ### First group: sides 25 ft and 34 ft 3. Calculate the range for the third side: $$|25 - 34| = 9$$ $$25 + 34 = 59$$ So the third side $c$ must satisfy: $$9 < c < 59$$ 4. Check each option: - a) 8 ft: $8 < 9$ (No) - b) 10 ft: $9 < 10 < 59$ (Yes) - c) 59 ft: $c$ must be less than 59, so 59 is not valid (No) - d) 61 ft: $61 > 59$ (No) --- ### Second group: sides 30 cm and 34 cm 5. Calculate the range for the third side: $$|30 - 34| = 4$$ $$30 + 34 = 64$$ So the third side $c$ must satisfy: $$4 < c < 64$$ 6. Check each option: - 5 cm: $4 < 5 < 64$ (Yes) - 18 cm: $4 < 18 < 64$ (Yes) - 29 cm: $4 < 29 < 64$ (Yes) - 50 cm: $4 < 50 < 64$ (Yes) - 65 cm: $65 > 64$ (No) - 72 cm: $72 > 64$ (No) --- **Final answers:** - First group: 10 ft - Second group: 5 cm, 18 cm, 29 cm, 50 cm