Question

the lengths of two sides of a triangle are 6 and 8, then the third side could be: (select all that apply)
a 7
b 2
c 14
d 15
e 2.5
f 14
g 10

Explanation:

Step1: Recall triangle inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(|a - b| < c < a + b\). Let \(a = 6\) and \(b = 8\). First, calculate \(a + b\) and \(|a - b|\).
\(a + b=6 + 8 = 14\)
\(|a - b|=|6 - 8| = 2\)
So the third side \(c\) must satisfy \(2 < c < 14\).

Step2: Check each option

• Option A: \(7\). Since \(2<7<14\), it is valid.
• Option B: \(2\). But \(c\) must be greater than \(2\), so it is invalid.
• Option C: \(14\). \(c\) must be less than \(14\), so it is invalid.
• Option D: \(15\). \(15>14\), so it is invalid.
• Option E: \(2.5\). Since \(2<2.5<14\), it is valid.
• Option F: \(14\). \(c\) must be less than \(14\), so it is invalid.
• Option G: \(10\). Since \(2<10<14\), it is valid.

Answer:

A. 7, E. 2.5, G. 10