Question

Apply the Triangle Inequality Theorem to find the range of measures for the third side of a triangle with the first two sides equal to 24 and 30.(1 point)
Responses

6<s<54
6 less than s less than 54

6>s>54
6 greater than s greater than 54

−6<s<54
negative 6 less than s less than 54

−6<s<6

Answer 1

To apply the Triangle Inequality Theorem, we need to consider the inequalities that must hold for any triangle with sides \(a\), \(b\), and \(c\). According to the theorem, the following must be true for any triangle:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

In this case, we have two sides \(a = 24\) and \(b = 30\). Let \(c\) be the length of the third side, which we will denote as \(s\).

Now let's apply each part of the theorem:

1. \(24 + 30 > s\)
\(54 > s\)
This simplifies to \(s < 54\).

2. \(24 + s > 30\)
\(s > 30 - 24\)
\(s > 6\).

3. \(30 + s > 24\)
\(s > 24 - 30\)
\(s > -6\).
Since this condition will always hold true for positive \(s\), we focus on the more restrictive condition \(s > 6\).

Combining the useful inequalities \(s > 6\) and \(s < 54\), we get:

\[6 < s < 54\]

Thus, the correct range for the measure of the third side \(s\) is:

**6 < s < 54**