Question

Question
The measures of two sides of a triangle are 5.2 and 8.1. Determine the range of values for the third side of the triangle.(1 point)
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Answer 1

To determine the range of values for the length of the third side of a triangle, we can use the triangle inequality theorem, which states:

1. The sum of the lengths of any two sides must be greater than the length of the third side.

Let \( a = 5.2 \), \( b = 8.1 \), and \( c \) be the length of the third side. The triangle inequality gives us the following three inequalities:

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

Substituting the known values into these inequalities:

1. From \( 5.2 + 8.1 > c \):
\[
13.3 > c \quad \text{(or } c < 13.3\text{)}
\]

2. From \( 5.2 + c > 8.1 \):
\[
c > 8.1 - 5.2
\]
\[
c > 2.9
\]

3. From \( 8.1 + c > 5.2 \):
\[
c > 5.2 - 8.1
\]
\[
c > -2.9 \quad \text{(This doesn't provide a new restriction since } c \text{ must be positive)}
\]

Combining the results from the inequalities, we find that:

\[
2.9 < c < 13.3
\]

So, the range of values for the third side \( c \) is:

\[
2.9 < c < 13.3
\]

In answer-box format:
\[
2.9 < n < 13.3
\]