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

5.5
5.5

4.5
4.5

4.2
4.2

3.5
3.5

Answer 1

To determine a possible measure for the third side of a triangle given the two sides (3.7 and 8.2), 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.

Let's denote the lengths of the two known sides as \( a = 3.7 \) and \( b = 8.2 \), and the unknown third side as \( c \).

According to the Triangle Inequality Theorem, we have the following conditions:

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

Now, we will derive the inequalities for \( c \):

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

The first two inequalities give us the most relevant information about \( c \):

- \( c < 11.9 \)
- \( c > 4.5 \)

Combining these inequalities means:

\[
4.5 < c < 11.9
\]

Therefore, the possible lengths for \( c \) must be greater than 4.5. Among the provided responses, the only value that satisfies this inequality is:

**5.5**

So the correct response is **5.5**.