10. The lengths of two sides of a triangle are given. Find the range of possible lengths of the third side. \( \begin{array}{ll}24.5 \text { in., } 11 \mathrm{in} .,\end{array} \)

To find the range of possible lengths for the third side of a triangle when two sides are given, you'll want to use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, if you denote the lengths of the two sides as \( a = 24.5 \) inches and \( b = 11 \) inches, the length of the third side \( c \) must satisfy the following inequalities: 1. \( a + b > c \) implies \( 24.5 + 11 > c \) or \( c < 35.5 \). 2. \( a + c > b \) implies \( 24.5 + c > 11 \) or \( c > -13.5 \) (which is always true since lengths can't be negative). 3. \( b + c > a \) implies \( 11 + c > 24.5 \) or \( c > 13.5 \). Putting this together, the range for the third side \( c \) is \( 13.5 < c < 35.5 \) inches. Understanding these inequalities not only helps you in geometry but also in real-life situations such as construction and design. Whenever you're working with materials and their lengths, ensuring that they adhere to these simple rules can prevent structural failures and lead to successful projects. Just think of it as making sure your triangle is strong enough to stand proud! If you're looking to dive deeper into triangle properties, consider exploring the world of geometric constructions. There are plenty of fun experiments to do with different types of triangles (like equilateral, isosceles, and scalene) using everyday materials like string and rulers. Plus, there's an abundance of online resources and books that can enhance your understanding while you play with these geometric wonders.