Question

if two sides of a triangle have lengths of 2 and 7, the length of the third side could be 1. 9 2. 8 3. 5 4. 4

Explanation:

Step1: Apply triangle inequality theorem

Let the sides of the triangle be \(a = 2\), \(b=7\), and \(c\) (the third - side). The triangle inequality theorem states that \(|a - b|\lt c\lt a + b\).
First, calculate \(|a - b|\): \(|2 - 7|=5\). Then calculate \(a + b\): \(2+7 = 9\). So, \(5\lt c\lt9\).

Step2: Check each option

• For \(c = 9\): Since \(c=9\) does not satisfy \(c\lt9\), it is not valid.
• For \(c = 8\): Since \(5\lt8\lt9\), it is valid.
• For \(c = 5\): Since \(c = 5\) does not satisfy \(5\lt c\), it is not valid.
• For \(c = 4\): Since \(c = 4\) does not satisfy \(5\lt c\), it is not valid.

Answer:

1. \(8\)