Question

Apply the triangle inequality theorem to define the range of measures for the third side of a triangle with the first two sides equal to 24 and 30
6<s<54
6>s>54
-6<s<54
-6< s<6

Answer 1

To apply the triangle inequality theorem to find the range of measures for the third side of a triangle, we have two sides with lengths 24 and 30. 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 third side.

Let the length of the third side be \(s\). According to the triangle inequality, we have the following three inequalities:

1. \( 24 + 30 > s \)
2. \( 24 + s > 30 \)
3. \( 30 + s > 24 \)

Let's solve these inequalities one by one:

1. From \( 24 + 30 > s \):
\[
54 > s \quad \text{or} \quad s < 54
\]

2. From \( 24 + s > 30 \):
\[
s > 30 - 24 \quad \Rightarrow \quad s > 6
\]

3. From \( 30 + s > 24 \):
\[
s > 24 - 30 \quad \Rightarrow \quad s > -6
\]
(This last inequality is always satisfied for positive \(s\), so we can ignore it for the range).

Combining the valid inequalities, we get:
\[
6 < s < 54
\]

Thus, the appropriate range for the length of the third side \(s\) is **\(6 < s < 54\)**.