Question

if 2 side of a triangle are 6 and 16, what is the range of the possible lengths for the third side?

Answer 1

To determine the possible lengths for the third side of a triangle when the lengths of the other two sides are given (6 and 16 in this case), we can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let \( a = 6 \), \( b = 16 \), and \( c \) be the length of the third side. The Triangle Inequality gives us the following three inequalities:

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

Plugging in the values we have:

1. \( 6 + 16 > c \) ⟹ \( 22 > c \) ⟹ \( c < 22 \)
2. \( 6 + c > 16 \) ⟹ \( c > 10 \)
3. \( 16 + c > 6 \) ⟹ This inequality is always true for positive \( c \).

Combining the constraints from the first two inequalities, we get:

\[
10 < c < 22
\]

Thus, the range of possible lengths for the third side \( c \) is \( (10, 22) \).