Question

1. if a triangle has sides that measure 8cm and 12cm, which of the following could be the third side? (2 points)

○ 3 cm
○ 20 cm
○ 5 cm
○ 24 cm

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. Mathematically, for a triangle with sides \(a\), \(b\), and \(c\), we have \(|a - b| < c < a + b\).

Here, \(a = 8\) cm and \(b = 12\) cm. First, calculate the difference and the sum:

• Difference: \(|12 - 8| = 4\) cm
• Sum: \(12 + 8 = 20\) cm

So, the third side \(c\) must satisfy \(4 < c < 20\).

Step2: Check each option

• For \(3\) cm: \(3 < 4\), so it does not satisfy the inequality.
• For \(20\) cm: \(20\) is not less than \(20\) (it's equal), so it does not satisfy the inequality.
• For \(5\) cm: \(4 < 5 < 20\), so it satisfies the inequality.
• For \(24\) cm: \(24 > 20\), so it does not satisfy the inequality.

Answer:

5 cm (the option corresponding to 5 cm)