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

4.2
4.2

3.5
3.5

5.5

:)

Answer 1

Good job

Answer 2

To apply the Triangle Inequality Theorem, we need to remember 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. \(b + c > a\)

Let the two sides of the triangle be \(a = 3.7\) and \(b = 8.2\), and the third side be \(c\).

Now we will apply the Triangle Inequality Theorem:

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

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

3. For \(b + c > a\):
\[
8.2 + c > 3.7 \implies c > 3.7 - 8.2 \implies c > -4.5
\]
(This inequality is trivial since \(c\) must be positive.)

Now, combining the results, we have:
\[
4.5 < c < 11.9
\]

From the provided options, \(4.5\) is not a valid measure (since it cannot be equal). Therefore, the possible measures for \(c\) are any value greater than \(4.5\) but less than \(11.9\).

The following options are given:
- 4.5
- 4.2
- 3.5
- 5.5

Only \(5.5\) satisfies \(c > 4.5\).

So, the possible measure of the third side is **5.5**.