Question

Is angle $\mathrm{A}$ the largest and side a the longest? (1) The sum of the length of the other two sides is greater than the length of a (2) A is the largest angle A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

B. 2 alone, not 1 alone

Step-by-step solution

Understanding the problem

Determine if angle $\text{A}$ is the largest angle and if side $a$ is the longest side based on the given conditions.

Analyzing statement 1

Statement 1 says that the sum of the lengths of the other two sides is greater than the length of $a$. This implies that $a$ could potentially be the longest side based on the triangle inequality theorem, which applies to all triangles, but does not guarantee that $a$ is the longest side nor that $\text{A}$ is the largest angle. Thus, statement 1 alone is insufficient.

Analyzing statement 2

Statement 2 says that angle $\text{A}$ is the largest angle in the triangle. According to the properties of triangles, the side opposite the largest angle is the longest side. Therefore, if $\text{A}$ is the largest angle, side $a$ must be the longest side. Hence, statement 2 alone is sufficient to answer the question.

Considering both statements together

Since statement 2 alone is sufficient to determine that side $a$ is the longest and angle $\text{A}$ is the largest, analyzing both statements together is not necessary.

Conclusion

Based on the analysis, statement 2 alone is sufficient to answer the question. Statement 1 alone is insufficient, and both statements together do not provide any additional information.

Key concepts

Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental concept in geometry. It states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem is a necessary condition for a set of three lines to form a triangle.

For example, if we have a triangle with sides of lengths $a$, $b$, and $c$, the following must always hold true:

• $a+b>c$
• $a+c>b$
• $b+c>a$

In the given problem, statement 1 refers to this theorem by saying that the sum of the lengths of the other two sides is greater than the length of side $a$. However, this alone is not sufficient to determine whether side $a$ is the longest or whether angle $A$ is the largest because this condition applies to all triangles.

Largest Angle in a Triangle

In a triangle, the largest angle is always opposite the longest side. This means that if you know which angle is the largest, you can directly infer which side is the longest.

Therefore, if it is stated that angle $A$ is the largest angle in the triangle, it must be opposite the longest side. By this property, we can conclude that side $a$ is the longest in the triangle when angle $A$ is the largest.

In our problem, statement 2 states that angle $A$ is the largest, leading us to conclude that side $a$ opposite to it is the longest. This is sufficient to answer the question, making statement 2 alone sufficient.

Longest Side of a Triangle

The longest side of a triangle is always opposite the largest angle, following the above concept. This is a straightforward way to determine relationships within triangle geometry.

If you identify the longest side, you can also identify the largest angle based on its opposite position. This property helps solve many triangle problems efficiently.

In the given exercise, although the sum of the other two sides being greater than side $a$ (statement 1) is a necessary condition, it doesn't help directly pinpoint the longest side or the largest angle. However, knowing that angle $A$ is the largest (statement 2) directly gives us this information.

Sufficient Conditions in GMAT Questions

GMAT problem solving often involves determining whether given statements or conditions are sufficient to answer a question. A statement is sufficient if it provides enough information to derive a conclusive answer without needing additional data.

In the problem provided, we're given two statements:

• Statement 1: The sum of the lengths of the other two sides is greater than $a$.
• Statement 2: Angle $A$ is the largest angle.

We evaluate each statement to decide if they individually or collectively are sufficient.

In this case, statement 1 alone is not sufficient because it doesn't ensure that side $a$ is the longest or that angle $A$ is the largest. But statement 2 alone is sufficient because it directly indicates that side $a$ is the longest due to being opposite the largest angle. Hence, the problem is solved, and option B (statement 2 alone) is correct.