Question

1. without using the law of cosines, explain why it is impossible to construct a triangle with lengths 9.6 cm, 12.1 cm, and 23.2 cm.

Explanation:

Step1: Recall triangle - side relationship

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Step2: Check the sum of two shorter sides

The two shorter sides are \(a = 9.6\space cm\) and \(b=12.1\space cm\). Calculate \(a + b\):
\(a + b=9.6+12.1 = 21.7\space cm\)

Step3: Compare with the longest side

The longest side \(c = 23.2\space cm\). Since \(21.7<23.2\) (i.e., \(a + b

Answer:

It is impossible to construct a triangle with side lengths \(9.6\space cm\), \(12.1\space cm\), and \(23.2\space cm\) because the sum of the two shorter sides (\(9.6 + 12.1=21.7\space cm\)) is less than the length of the longest side (\(23.2\space cm\)), violating the triangle - side relationship theorem (the sum of the lengths of any two sides of a triangle must be greater than the length of the third side).