Question

1. the lengths of two sides of a triangle are given. find the range of possible lengths for the third side

a) 5 in, 16 in

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that the length of any side of a triangle must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. For a triangle with sides \(a\), \(b\), and \(c\), we have \(|a - b| < c < a + b\).

Step2: Identify the given sides

Here, the two given sides are \(a = 5\) in and \(b = 16\) in.

Step3: Calculate the difference and sum

First, find the absolute difference: \(|16 - 5| = 11\).
Then, find the sum: \(16 + 5 = 21\).

Step4: Determine the range for the third side

Using the triangle inequality theorem, the length of the third side \(c\) must satisfy \(11 < c < 21\).

Answer:

The range of possible lengths for the third side is \(11\) inches \(<\) third side \(< 21\) inches.