Questions: A triangle has two sides of length 9 and 2. What is the largest possible whole number length for the third side?

Solution Steps

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This applies to all three combinations of sides.

Let the sides of the triangle be \( a = 9 \), \( b = 2 \), and \( c \) (the unknown side). According to the theorem:

1. \( a + b > c \) → \( 9 + 2 > c \) → \( 11 > c \).
2. \( a + c > b \) → \( 9 + c > 2 \) → \( c > -7 \). Since lengths are positive, this condition is always satisfied.
3. \( b + c > a \) → \( 2 + c > 9 \) → \( c > 7 \).

From the inequalities:

• \( c < 11 \).
• \( c > 7 \).

Thus, \( c \) must satisfy \( 7 < c < 11 \).

The largest whole number that satisfies \( 7 < c < 11 \) is \( 10 \).

Final Answer