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

3.5
3.5

4.2

Answer 1

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For sides measuring 3.7 and 8.2, let \( x \) represent the length of the third side. We can create the following inequalities based on the theorem:

1. \( 3.7 + 8.2 > x \)
2. \( 3.7 + x > 8.2 \)
3. \( 8.2 + x > 3.7 \)

Now we can solve each inequality:

1. \( 3.7 + 8.2 > x \)
\[
11.9 > x \quad \text{or} \quad x < 11.9
\]

2. \( 3.7 + x > 8.2 \)
\[
x > 8.2 - 3.7
\]
\[
x > 4.5
\]

3. \( 8.2 + x > 3.7 \)
\[
x > 3.7 - 8.2
\]
\[
x > -4.5 \quad \text{(this inequality does not provide new information)}
\]

Combining the relevant inequalities, we have:
\[
4.5 < x < 11.9
\]

From the provided options (5.5, 4.5, 3.5, 4.2), the possible measures for the third side that satisfy the Triangle Inequality Theorem are:

- 5.5 (since \( 4.5 < 5.5 < 11.9 \))

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