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

4.2
4.2

3.5
3.5

4.5
4.5

5.5

Answer 1

To find a possible measure of the third side of a triangle with sides measuring 3.7 and 8.2, we can use the Triangle Inequality Theorem. The theorem states that for any triangle with sides \( a \), \( b \), and \( c \):

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

Here, we can set \( a = 3.7 \) and \( b = 8.2 \), and let \( c \) be the unknown side we want to find.

Applying the Triangle Inequality Theorem:

1. \( 3.7 + 8.2 > c \)
- \( 11.9 > c \) → \( c < 11.9 \)

2. \( 3.7 + c > 8.2 \)
- \( c > 8.2 - 3.7 \)
- \( c > 4.5 \)

3. \( 8.2 + c > 3.7 \)
- \( c > 3.7 - 8.2 \)
- Since \( 3.7 - 8.2 \) is negative, this does not provide a new restriction.

From these inequalities, we find that \( 4.5 < c < 11.9 \).

Based on the provided options:
- 4.2: Not possible because \( c > 4.5 \)
- 3.5: Not possible because \( c > 4.5 \)
- 4.5: Not possible because \( c > 4.5 \)
- 5.5: Possible because \( 4.5 < 5.5 < 11.9 \)

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