Apply the Triangle Inequality Theorem to determine the possible whole number measures of the third side of a triangle if the first two sides measure 6 and 2. List them in ascending order.(2 points)

Question

The measure of the third side could be ___, ___, or___ .

Answer 1

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

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

In this case, let’s denote the sides as follows:

• \( a = 6 \)
• \( b = 2 \)
• \( c \) is the third side we want to determine.

Let's apply the Triangle Inequality Theorem:

1. From \( a + b > c \): \[ 6 + 2 > c \implies 8 > c \implies c < 8 \]

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

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

Now, combining the valid inequalities for \( c \):

• From \( c < 8 \)
• From \( c > 4 \)

Therefore, \( c \) must satisfy: \[ 4 < c < 8 \]

The possible whole number values for \( c \) are 5, 6, and 7.

In ascending order, the measure of the third side could be 5, 6, or 7.