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

3.5
3.5

4.5
4.5

4.2
4.2

5.5
5.5

Answer 1

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

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

Given two sides of the triangle, 3.7 and 8.2, let's denote them as \(a = 3.7\) and \(b = 8.2\). We can find a range of possible values for the third side \(c\) using the triangle inequality:

1. From \(a + b > c\):
\[
3.7 + 8.2 > c \implies 11.9 > c \implies c < 11.9
\]

2. From \(a + c > b\):
\[
3.7 + c > 8.2 \implies c > 8.2 - 3.7 \implies c > 4.5
\]

3. From \(b + c > a\):
\[
8.2 + c > 3.7 \implies c > 3.7 - 8.2 \implies c > -4.5 \quad \text{(this inequality does not constrain \(c\))}
\]

Combining the relevant inequalities, we find that:
\[
4.5 < c < 11.9
\]

Now, let's examine the provided options for possible measures of the third side:
- **3.5**: does not satisfy \(c > 4.5\).
- **4.5**: does not satisfy \(c > 4.5\) (it is equal, not greater).
- **4.2**: does not satisfy \(c > 4.5\).
- **5.5**: satisfies \(4.5 < c < 11.9\).

Therefore, the only valid possible measure of the third side is **5.5**.