Without drawing, how can you show if it is possible to form a triangle with side lengths 5, 7, and 10? Explain why your method works.

To determine whether the lengths 5, 7, and 10 can form a triangle without drawing it, you can use the **Triangle Inequality Theorem**. This theorem states that for any three lengths to form a valid triangle, the sum of the lengths of **any two sides** must be **greater than** the length of the remaining side. Here's how you can apply this method: 1. **Check the first pair:** - **5 + 7 > 10** - **12 > 10** ✔️ 2. **Check the second pair:** - **5 + 10 > 7** - **15 > 7** ✔️ 3. **Check the third pair:** - **7 + 10 > 5** - **17 > 5** ✔️ Since all three conditions are satisfied, the lengths 5, 7, and 10 **can** form a valid triangle. **Why This Method Works:** The Triangle Inequality Theorem ensures that the sides are long enough to "connect" and close the shape into a triangle. If any one of these conditions fails (for example, if 5 + 7 were not greater than 10), the sides wouldn't meet properly to form a closed three-sided shape. This theorem is fundamental in geometry for validating the possibility of forming a triangle with given side lengths.