Question:medium

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: \( \angle ABC = 30^{\circ} \)

Show Hint

Use the geometric-progression condition or the law of cosines with the given angle to fix AC.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is D

Solution and Explanation

Step 1: Plan a coordinate approach for one statement and a case check for the other.
Placing the triangle on axes turns statement 2 into a distance calculation, while statement 1 is best handled by checking which arrangement of the progression survives the triangle inequality.

Step 2: Handle statement 2 with coordinates.
Put B at the origin and A at (2, 0), along the positive x-axis.
Since angle ABC = 30 degrees, point C sits at distance 4 from B along a ray 30 degrees from BA, so \( C = (4\cos30^{\circ}, 4\sin30^{\circ}) = (2\sqrt{3}, 2) \).
The distance AC is \( \sqrt{(2\sqrt{3}-2)^{2} + 2^{2}} = \sqrt{(16 - 8\sqrt{3}) + 4} = \sqrt{20 - 8\sqrt{3}} \) cm.
This single number matches the law-of-cosines result, so statement 2 alone is sufficient.

Step 3: Handle statement 1 by checking placements.
A geometric progression built from 2 and 4 could place the third term before 2, between 2 and 4, or after 4.
Placing AC after 4 or before 2 forces side lengths like 2, 4, 8, which fail the triangle inequality since \( 2 + 4 \) is not greater than 8.
Only placing AC between 2 and 4 survives, giving \( AC^{2} = 2 \times 4 = 8 \), so \( AC = 2\sqrt{2} \) cm.
Since only one placement is geometrically valid, statement 1 alone also gives a single fixed length.

Final Answer:
Coordinates confirm statement 2 works alone, and the placement check confirms statement 1 works alone. \[ \boxed{(d)} \]
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