Two sides of a triangle are 6 in and 8 in. Using the Triangle Inequality Theorem (see picture above), which amount could be the length of the third side from the choices given?

Question

(1 point)
Responses

10 in.
10 in.

2 in.
2 in.

1 in.
1 in.
Skip to navigation

Answer 1

To determine the possible lengths for the third side of a triangle using the Triangle Inequality Theorem, we need to apply the theorem's three inequalities. Given two sides of lengths 6 in and 8 in, let \( x \) be the length of the third side. The Triangle Inequality Theorem states:

1. \( 6 + 8 > x \)
2. \( 6 + x > 8 \)
3. \( 8 + x > 6 \)

Now, we can apply these inequalities:

1. \( 14 > x \) or \( x < 14 \)
2. \( 6 + x > 8 \) simplifies to \( x > 2 \)
3. \( 8 + x > 6 \) simplifies to \( x > -2 \) (which is always true for a positive length)

Combining the relevant results from the inequalities, we find:

• \( x < 14 \)
• \( x > 2 \)

Thus, the possible range for the third side \( x \) is: \[ 2 < x < 14 \]

From the provided choices:

• 10 in. (valid, since \( 2 < 10 < 14 \))
• 2 in. (invalid, not greater than 2)
• 1 in. (invalid, not greater than 2)

The valid choice for the length of the third side is 10 in.