Question

Two side of a triangle measures 4cm and 9 cm which of the following choices cannot be the measure of the third side

Answer 1

To determine the possible measure of the third side of a triangle when two sides are given, we can use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given the two sides of the triangle as \(a = 4 \, \text{cm}\) and \(b = 9 \, \text{cm}\), we denote the length of the third side as \(c\). The Triangle Inequality Theorem gives us the following inequalities:

1. \(a + b > c\) (which means \(4 + 9 > c\) or \(c < 13\))
2. \(a + c > b\) (which means \(4 + c > 9\) or \(c > 5\))
3. \(b + c > a\) (which means \(9 + c > 4\) or \(c > -5\), which is always true since side lengths are positive)

From the first and second inequalities, we can summarize:

- \(c < 13\)
- \(c > 5\)

Thus, the measure of the third side \(c\) must satisfy:

\[ 5 < c < 13 \]

Now, any value of \(c\) that falls outside of this range cannot be the measure of the third side. Therefore, if you have specific choices in mind, any option that is \( \leq 5\) or \( \geq 13\) cannot be the measure of the third side.