Question Two sides of a right triangle have lengths of 5 inches and 12 inches. Which of the following could be the length, in inches, of the third side of the triangle?

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1. There are two possible cases for how the given sides can fit into a right triangle:
   - **Case 1:** Both $$5$$ inches and $$12$$ inches are the legs.
   - **Case 2:** One of them is the hypotenuse.
   
2. **Case 1: $$5$$ inches and $$12$$ inches are the legs**  
   Apply the Pythagorean theorem:
   $$
   5^2 + 12^2 = c^2
   $$
   where $$c$$ is the hypotenuse. Compute the squares:
   $$
   25 + 144 = c^2 \quad\Longrightarrow\quad 169 = c^2
   $$
   Taking the square root:
   $$
   c = 13
   $$
   So, one possible third side is $$13$$ inches.
   
3. **Case 2: One of the given sides is the hypotenuse**  
   The hypotenuse must be the longest side, so only $$12$$ inches could be the hypotenuse (since $$12 > 5$$). Let $$x$$ be the unknown leg. Then:
   $$
   5^2 + x^2 = 12^2
   $$
   Compute the squares:
   $$
   25 + x^2 = 144
   $$
   Solve for $$x^2$$:
   $$
   x^2 = 144 - 25 = 119
   $$
   Taking the square root:
   $$
   x = \sqrt{119}
   $$
   Thus, the third side could also be $$\sqrt{119}$$ inches (approximately $$10.91$$ inches).

4. **Conclusion:**  
   The third side of the right triangle could be either $$13$$ inches (if the $$5$$ and $$12$$ inches are legs) or $$\sqrt{119}$$ inches (if $$12$$ inches is the hypotenuse).
The third side of the right triangle could be either 13 inches or approximately 10.91 inches.

Question Two sides of a right triangle have lengths of 5 inches and 12 inches. Which of the following could be the length, in inches, of the third side of the triangle?

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