1. **State the problem:** Determine if a triangle can have side lengths 3, 7, and 10. 2. **Recall the triangle inequality theorem:** For any triangle with sides $a$, $b$, and $c$, the sum of the lengths of any two sides must be greater than the length of the remaining side. That is: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ 3. **Apply the theorem to the given sides:** - Check $3 + 7 > 10$: $10 > 10$ (False, since it is equal, not greater) - Check $3 + 10 > 7$: $13 > 7$ (True) - Check $7 + 10 > 3$: $17 > 3$ (True) 4. **Conclusion:** Since $3 + 7$ is not greater than $10$, the triangle inequality is not satisfied. Therefore, a triangle with sides 3, 7, and 10 cannot exist.