Question

Apply the Triangle Inequality Theorem and determine which option could be the third side of a triangle if the first two sides measure 4 and 6.(1 point) Responses 9 9 2 2 11 11 1

Answer 1

To determine the possible lengths of the third side of a triangle given the lengths of the first two sides, we can apply the Triangle Inequality Theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

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

In this case, let's call the two known sides \( a = 4 \) and \( b = 6 \), and let \( c \) be the length of the third side.

Applying the Triangle Inequality Theorem:

1. From \( a + b > c \):
\[
4 + 6 > c \implies 10 > c \implies c < 10
\]

2. From \( a + c > b \):
\[
4 + c > 6 \implies c > 2
\]

3. From \( b + c > a \):
\[
6 + c > 4 \implies c > -2 \text{ (this is always true since c is positive)}
\]

From the inequalities we deduced, we can conclude that:
\[
2 < c < 10
\]

Now, let’s evaluate each option to see which values fall within the range \( 2 < c < 10 \):
- 9 (valid, since \( 2 < 9 < 10 \))
- 2 (not valid, since \( 2 \) is not greater than \( 2 \))
- 11 (not valid, since \( 11 \) is not less than \( 10 \))
- 1 (not valid, since \( 1 < 2 \))

Thus, the only valid option for the length of the third side \( c \) satisfying the Triangle Inequality Theorem is **9**.