Question

Two sides of triangle measure 3.7 and 8.2 what is the possible measure of the third side?
4.5
4.2
3.5
5.5

Answer 1

To determine the possible measure of the third side of a triangle given two sides, you can use the triangle inequality theorem, which states:

1. The sum of the lengths of any two sides must be greater than the length of the third side.

Given the two sides measuring 3.7 and 8.2, let \( c \) be the length of the third side. The triangle inequalities can be expressed as follows:

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

Let's evaluate these inequalities one by one:

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

2. From \( 3.7 + c > 8.2 \):
\[
c > 8.2 - 3.7 \quad \text{which simplifies to} \quad c > 4.5
\]

3. From \( 8.2 + c > 3.7 \):
\[
c > 3.7 - 8.2 \quad \text{which simplifies to} \quad c > -4.5 \quad \text{(this inequality is always true for positive lengths)}
\]

Thus, combining the meaningful inequalities, we have:
\[
4.5 < c < 11.9
\]

Now, given the options: 4.5, 4.2, 3.5, and 5.5, we can see that:
- 4.5: does not satisfy \( c > 4.5 \)
- 4.2: does not satisfy \( c > 4.5 \)
- 3.5: does not satisfy \( c > 4.5 \)
- 5.5: satisfies \( 4.5 < 5.5 < 11.9 \)

The only viable option for the measure of the third side is **5.5**.