A triangle has two sides of length 2 and 5. 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 need to 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.

2. Relevant Concepts

According to the triangle inequality theorem, for sides of lengths $a$, $b$, and $c$:

• $a + b > c$
• $a + c > b$
• $b + c > a$

In this problem, we can set:

• $a = 2$
• $b = 5$
• $c$ as the unknown length of the third side.

3. Analysis and Detail

We can rearrange the inequalities defined by the triangle inequality theorem for this specific case:

1. $2 + 5 > c$

   • This simplifies to: $c < 7$

2. $2 + c > 5$

   • This simplifies to: $c > 3$

3. $5 + c > 2$

   • This simplifies to: $c > -3 \quad \text{(always true, thus irrelevant here)}$

From the inequalities, we have:

• $3 < c < 7$

Therefore, the possible whole-number lengths for $c$ would be 4, 5, or 6.

4. Verify and Summarize

Among the whole numbers satisfying our inequalities, the largest possible length for the third side is 6.

Final Answer

The largest possible whole-number length for the third side is $\boxed{6}$.