A triangle has two sides of length 16 and 4. What is the largest possible whole-number length for the third side?

1. ### Break Down the Problem

To find the largest possible whole-number length for the third side of a triangle, we can use the triangle inequality theorem. This theorem states that the sum of any two sides of a triangle must be greater than the third side.

2. ### Relevant Concepts

Let the lengths of the sides of the triangle be $a = 16$, $b = 4$, and $c$ (the third side). According to the triangle inequality theorem, we need to satisfy the following conditions:

1. $a + b > c$
2. $a + c > b$
3. $b + c > a$

3. ### Analysis and Detail

We will analyze each of these conditions:

1. From $a + b > c$: $16 + 4 > c \implies 20 > c \implies c < 20$

2. From $a + c > b$: $16 + c > 4 \implies c > 4 - 16 \implies c > -12 \quad (\text{This condition is always satisfied since } c > 0)$

3. From $b + c > a$: $4 + c > 16 \implies c > 16 - 4 \implies c > 12$

From the inequalities above, we have:

• $c < 20$
• $c > 12$

4. ### Verify and Summarize

Putting these two inequalities together, we find: $12 < c < 20$

This means that $c$ can take any whole number value from 13 to 19. The largest possible whole-number length for $c$ is 19.

Final Answer

The largest possible whole-number length for the third side is 19.