Question

the lengths of two sides of a triangle are 5 inches and 8 inches. which of the following lengths could be the length of the third side of the triangle? select all that apply. 4 inches 5 inches 7 inches 13 inches

Explanation:

Step1: Recall triangle inequality theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side, and the difference of the lengths of any two sides must be less than the length of the remaining side. For a triangle with sides \(a\), \(b\), and \(c\), we have \(|a - b| < c < a + b\). Here, \(a = 5\) inches and \(b = 8\) inches. So, first, calculate the range of the third side \(c\).

Step2: Calculate the range of the third side

First, find the difference: \(|8 - 5|=3\) inches. Then, find the sum: \(8 + 5 = 13\) inches. So the third side \(c\) must satisfy \(3 < c < 13\).

Step3: Check each option

• For 4 inches: \(3 < 4 < 13\), so it is valid.
• For 5 inches: \(3 < 5 < 13\), so it is valid.
• For 7 inches: \(3 < 7 < 13\), so it is valid.
• For 13 inches: \(13\) is not less than \(13\) (since \(c < 13\)), so it is not valid.

Answer:

4 inches, 5 inches, 7 inches (the checkboxes corresponding to these lengths should be selected)