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 possible range of values for the third side \( n \) of a triangle with sides of lengths 5.2 and 8.1, we can use the triangle inequality theorem. This 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.

We can express this in three inequalities:

1. \( 5.2 + 8.1 > n \)
2. \( 5.2 + n > 8.1 \)
3. \( 8.1 + n > 5.2 \)

Now, let's solve each inequality:

1. **From the first inequality:**
\[
5.2 + 8.1 > n \implies 13.3 > n \implies n < 13.3
\]

2. **From the second inequality:**
\[
5.2 + n > 8.1 \implies n > 8.1 - 5.2 \implies n > 2.9
\]

3. **From the third inequality:**
\[
8.1 + n > 5.2 \implies n > 5.2 - 8.1 \implies n > -2.9
\]
(This inequality is always satisfied since the value for \( n \) must be positive.)

Now, we can combine the results from the first two inequalities:

Thus, the range of possible values for the third side \( n \) is:
\[
2.9 < n < 13.3
\]

Final answer:
\[
2.9 < n < 13.3
\]