Question

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 5.5 5.5 3.5


exsplain how

Answer 1

The Triangle Inequality Theorem states 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. \(b + c > a\)

In this case, we have two sides of the triangle measuring \(a = 3.7\) and \(b = 8.2\). We need to find a possible measure for the third side, which we will denote as \(c\).

We can apply the Triangle Inequality Theorem to find the possible values of \(c\):

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

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

3. From the inequality \(b + c > a\):
\[
8.2 + c > 3.7 \implies c > 3.7 - 8.2 \implies c > -4.5
\]
(This inequality is always true for positive values of \(c\).)

Now, combining the valid inequalities from the first two calculations:
- We have \(c < 11.9\)
- We have \(c > 4.5\)

So the possible range for \(c\) is:
\[
4.5 < c < 11.9
\]

From the options provided (4.5, 4.2, 5.5, 3.5), the only value that satisfies \(c > 4.5\) is:
- **5.5**

Thus, a possible measure for the third side \(c\) is **5.5**.