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

11
11

9
9

1
1

2

Answer 1

To determine the possible lengths of the third side of a triangle when the first two sides measure 4 and 6, you can use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the lengths of the two sides as \( a = 4 \) and \( b = 6 \), and the length of the third side as \( c \). The Triangle Inequality Theorem gives us the following inequalities:

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

Substituting the values we have:

1. \( 4 + 6 > c \) → \( 10 > c \) → \( c < 10 \)
2. \( 4 + c > 6 \) → \( c > 2 \)
3. \( 6 + c > 4 \) → \( c > -2 \) (This inequality is always true for positive \( c \))

Combining the inequalities from the first two conditions, we find:

\[ 2 < c < 10 \]

Now, let's evaluate each of the given options:

- **11**: Not valid (greater than 10)
- **9**: Valid (between 2 and 10)
- **1**: Not valid (less than 2)
- **2**: Not valid (equal to 2, must be greater than 2)

The only option that could be the length of the third side is **9**.