A triangle has two sides of length 19 and 20. What is the smallest possible whole-number length for the third side?Submit

To determine the smallest possible whole-number length for the third side of a triangle with sides of length 19 and 20, we can utilize 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.

1. ### Break Down the Problem

• Let $a = 19$, $b = 20$, and $c$ be the length of the third side.
• We need to find the smallest whole number $c$ such that the triangle inequality holds.

2. ### Relevant Concepts

According to the triangle inequality:

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

3. ### Analysis and Detail

Now we will apply these inequalities:

1. From the first inequality: $19 + 20 > c \implies 39 > c \implies c < 39$

2. From the second inequality: $19 + c > 20 \implies c > 20 - 19 \implies c > 1$

3. From the third inequality: $20 + c > 19 \implies c > 19 - 20 \implies c > -1 \text{ (This is always true)}$

From the relevant inequalities, we can conclude that: $c > 1 \quad \text{and} \quad c < 39$ Thus, for $c$ to be a whole-number, the smallest possible value satisfying these conditions is $c = 2$.

4. ### Verify and Summarize

All conditions for a valid triangle are satisfied with $c = 2$:

• $19 + 20 = 39 > 2$
• $19 + 2 = 21 > 20$
• $20 + 2 = 22 > 19$

Final Answer

The smallest possible whole-number length for the third side is $\boxed{2}$.