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

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

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To determine the range of possible lengths for the third side of a triangle when two sides are given, we can use the **Triangle Inequality Theorem**. This theorem states that for any triangle, the length of any one side must be: 1. **Less than the sum** of the lengths of the other two sides. 2. **Greater than the difference** of the lengths of the other two sides. Let's apply this to the given sides: - **Given Side 1 (a):** 24.5 inches - **Given Side 2 (b):** 11 inches - **Third Side (c):** ? ### Applying the Triangle Inequality Theorem 1. **Upper Bound (c < a + b):** $$ c < 24.5\ \text{in} + 11\ \text{in} = 35.5\ \text{in} $$ So, the third side must be **less than 35.5 inches**. 2. **Lower Bound (c > |a - b|):** $$ c > |24.5\ \text{in} - 11\ \text{in}| = 13.5\ \text{in} $$ So, the third side must be **greater than 13.5 inches**. ### Conclusion The length of the third side **must be greater than 13.5 inches** and **less than 35.5 inches**. **Therefore, the range of possible lengths for the third side is:** $$ 13.5\ \text{inches} < c < 35.5\ \text{inches} $$ The third side must be greater than 13.5 inches and less than 35.5 inches.

The lengths of two sides of a triangle are given. Find the range of possible lengths of the third side.

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