The lengths of two sides of a triangle are 7 cm and 17 cm .
Identify the range of possible lengths for the third side.

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The lengths of two sides of a triangle are 7 cm and 17 cm . Identify the range of possible lengths for the third side. $$ 10 \mathrm{~cm}

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To determine the range of possible lengths for the third side $$ x $$ of a triangle when two sides are 7 cm and 17 cm, we can use the **Triangle Inequality Theorem**. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Let's apply this theorem step by step: 1. **Sum of the two given sides must be greater than the third side:** $$ 7\, \text{cm} + 17\, \text{cm} > x \quad \Rightarrow \quad 24\, \text{cm} > x $$ 2. **Sum of the third side and one of the given sides must be greater than the other given side:** $$ x + 7\, \text{cm} > 17\, \text{cm} \quad \Rightarrow \quad x > 10\, \text{cm} $$ 3. **Sum of the third side and the other given side must be greater than the first given side:** $$ x + 17\, \text{cm} > 7\, \text{cm} \quad \Rightarrow \quad x > -10\, \text{cm} $$ *(This last inequality is always true for positive lengths, so it doesn't constrain $$ x $$ further.)* Combining the valid inequalities, we find: $$ 10\, \text{cm} < x < 24\, \text{cm} $$ **Therefore, the range of possible lengths for the third side is:** $$ 10\, \text{cm} < x < 24\, \text{cm} $$ **Answer:** $$ 10 \mathrm{~cm}The third side must be between 10 cm and 24 cm.

The lengths of two sides of a triangle are 7 cm and 17 cm .
Identify the range of possible lengths for the third side.

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