Question

In \(\triangle ABC\), if AB = a, BC = \(b+\frac{1}{2}\), and AC = b, which of the three angles of \(\triangle ABC\) has the greatest degree measure?
(1) \(b = a+\frac{1}{2}\)
(2) \(a = \frac{1}{2}\)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

In geometry data sufficiency, always remember foundational theorems. Here, the relationship between side lengths and opposite angles is key. Also, don't forget the Triangle Inequality Theorem. A statement is not sufficient if it contradicts the given information (e.g., that the figure is a triangle).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In any triangle, the angle with the greatest measure is opposite the longest side. To answer the question, we need to determine which of the three sides—AB, BC, or AC—is the longest. The lengths are given as \(a\), \(b+\frac{1}{2}\), and \(b\).
By simple inspection, BC (\(b+\frac{1}{2}\)) is longer than AC (\(b\)). So, the longest side is either BC or AB. The question reduces to comparing the lengths of AB (\(a\)) and BC (\(b+\frac{1}{2}\)).
The angles are \(\angle ACB\) (opposite AB), \(\angle BAC\) (opposite BC), and \(\angle ABC\) (opposite AC).
Step 2: Detailed Explanation:
Analyzing Statement (1):
This statement gives us the relation \(b = a+\frac{1}{2}\). We can rearrange this to express \(a\) in terms of \(b\): \(a = b - \frac{1}{2}\).
Now let's write the lengths of all three sides in terms of \(b\):
- AB = \(a = b - \frac{1}{2}\)
- BC = \(b + \frac{1}{2}\)
- AC = \(b\)
For these to form a valid triangle, we must have \(b>\frac{1}{2}\). Assuming a valid triangle exists, we can compare the side lengths:
\[ b + \frac{1}{2}>b>b - \frac{1}{2} \]
This means BC>AC>AB.
The longest side is BC. The angle opposite the longest side is the greatest angle. The angle opposite side BC is \(\angle BAC\).
Since we can uniquely identify the greatest angle, statement (1) alone is sufficient.
Analyzing Statement (2):
This statement says \(a = \frac{1}{2}\).
The side lengths are:
- AB = \(\frac{1}{2}\)
- BC = \(b + \frac{1}{2}\)
- AC = \(b\)
For these three lengths to form a triangle, the sum of any two sides must be greater than the third side (Triangle Inequality Theorem). Let's check this condition:
\[ \text{AB} + \text{AC}>\text{BC} \]
\[ \frac{1}{2} + b>b + \frac{1}{2} \]
This simplifies to \(b + \frac{1}{2}>b + \frac{1}{2}\), which is false. The sum of the two sides AB and AC is exactly equal to the length of the third side BC. This means the "triangle" is a degenerate triangle, where the vertices A, C, and B lie on a straight line.
Since the premise of the question "In \(\triangle ABC\)..." implies the existence of a non-degenerate triangle, this statement contradicts the premise. Therefore, it cannot be used to determine the greatest angle within a valid triangle. Statement (2) is not sufficient.
Step 3: Final Answer:
Statement (1) is sufficient to determine the longest side and thus the greatest angle. Statement (2) describes a situation where a valid triangle cannot be formed. Therefore, the correct option is (A).