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)
___< n <___

Answer 1

To determine the range of values for the third side of a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\):

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

In your case, let the two given sides be \(a = 5.2\) and \(b = 8.1\), and let \(c\) be the length of the third side.

We can apply the triangle inequality as follows:

1. \(5.2 + 8.1 > c\)
\[
13.3 > c \quad \Rightarrow \quad c < 13.3
\]

2. \(5.2 + c > 8.1\)
\[
c > 8.1 - 5.2 \quad \Rightarrow \quad c > 2.9
\]

3. \(8.1 + c > 5.2\)
\[
c > 5.2 - 8.1 \quad \Rightarrow \quad c > -2.9
\]
(This condition is always satisfied, as \(c\) is a positive length.)

Combining the relevant inequalities, we find:
\[
2.9 < c < 13.3
\]

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