Question

Two sides of a triangle measure 3.7 and 8.2. Apply the Triangle Inequality Theorem to find a possible measure of the third side.(1 point) Responses 4.5 4.5 3.5 3.5 5.5 5.5 4.2

Answer 1

The Triangle Inequality Theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following inequalities must hold:

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

In this case, let's denote the two sides of the triangle as \(a = 3.7\) and \(b = 8.2\), and let \(c\) be the length of the third side.

Using the Triangle Inequality Theorem:

1. **From the first inequality**:
\(3.7 + 8.2 > c\)
\(11.9 > c\)
or \(c < 11.9\)

2. **From the second inequality**:
\(3.7 + c > 8.2\)
\(c > 8.2 - 3.7\)
\(c > 4.5\)

3. **From the third inequality**:
\(8.2 + c > 3.7\)
\(c > 3.7 - 8.2\)
\(c > -4.5\) (This is always true for a positive side length, so we don't need this condition.)

Now we combine the results from the first two inequalities. The third side \(c\) must satisfy:

\[
4.5 < c < 11.9
\]

From the provided options (4.5, 3.5, 5.5, 4.2):

- **4.5** is not valid, as it must be greater than 4.5.
- **3.5** is not valid as it is less than 4.5.
- **5.5** is valid as it is between 4.5 and 11.9.
- **4.2** is not valid as it is less than 4.5.

Thus, a possible measure of the third side is **5.5**.