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

2
2

9
9

1
1

11
11

Answer 1

The Triangle Inequality Theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold true:

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

In your case, the lengths of the two sides are 4 and 6. Let's denote the length of the third side as \( c \). We can set up the inequalities as follows:

1. \( 4 + 6 > c \) → \( 10 > c \) or \( c < 10 \)
2. \( 4 + c > 6 \) → \( c > 2 \)
3. \( 6 + c > 4 \) → This will always be true since \( c \) is positive.

From these inequalities, we find that the third side \( c \) must satisfy \( 2 < c < 10 \).

Now we can check the provided options to see which ones fall within the range \( 2 < c < 10 \):

- **2**: Not valid since \( c \) must be greater than 2.
- **9**: Valid since it falls within the range.
- **1**: Not valid since \( c \) must be greater than 2.
- **11**: Not valid since \( c \) must be less than 10.

Thus, the only valid option for the third side is **9**.