Question

In \( \triangle ABC \), \( AB = 2 \) cm and \( BC = 4 \) cm. What is the length of AC?

Statement 1: The three sides of the triangle are in geometric progression
Statement 2: \( \angle ABC = 30^{\circ} \)

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Use the geometric-progression condition or the law of cosines with the given angle to fix AC.

Answer 1

The Correct Option is D

Step 1: Note what fixes AC.
With AB = 2 and BC = 4 already known, one more independent fact about the triangle should fix AC to a single value.

Step 2: Test statement 1 alone.
The sides are in geometric progression, and two of the three sides are 2 and 4.
If AC sat outside the pair 2 and 4 in the progression, the resulting side lengths would break the triangle inequality.
The only workable placement is AC as the middle term, so \( AC^{2} = AB \times BC = 2 \times 4 = 8 \), giving \( AC = 2\sqrt{2} \) cm.
Checking 2, \( 2\sqrt{2} \), 4 against the triangle inequality confirms a valid triangle.
This placement is forced, so statement 1 alone gives one fixed length.

Step 3: Test statement 2 alone.
Angle ABC sits between sides AB and BC, so the law of cosines applies directly.
\( AC^{2} = AB^{2} + BC^{2} - 2 \times AB \times BC \times \cos(30^{\circ}) = 4 + 16 - 16 \times \dfrac{\sqrt{3}}{2} = 20 - 8\sqrt{3} \).
This gives one positive value, \( AC = \sqrt{20 - 8\sqrt{3}} \) cm.
Statement 2 alone also gives one fixed length.

Final Answer:
Each statement alone is enough to fix the length of AC. \[ \boxed{(d)} \]