Question

In Δ ABC, AB = 2 cm and BC = 4 cm. What is the length of the AC?
Statement 1: The three sides of the triangle are in geometric progression
Statement 2: ∠ABC = 30°

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is D

To solve this problem, we need to determine the length of AC in triangle \( \triangle ABC \) with given sides AB = 2 cm and BC = 4 cm. Let's evaluate both statements individually and together to check their sufficiency.

• Statement 1: The three sides of the triangle are in geometric progression.
  A geometric progression means that the ratio between consecutive terms is constant. Let's denote the sides of the triangle as \( a, b, c \) (where \( a = \) AB, \( b = \) BC, and \( c = \) AC). Thus, we have the sides as \( 2, x, 4 \), where:
  • The ratio between first two sides: \( \frac{x}{2} \)
  • The ratio between last two sides: \( \frac{4}{x} \)
• Statement 2: \( \angle ABC = 30^\circ \).
  Given \( \angle ABC = 30^\circ \), we can apply the Law of Cosines, which is stated as: \(c^2 = a^2 + b^2 - 2ab \cos C\)
  Plugging in the known values \( a = 2, b = 4, C = 30^\circ \): \(c^2 = 2^2 + 4^2 - 2 \cdot 2 \cdot 4 \cdot \cos(30^\circ)\)
  \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
  \(c^2 = 4 + 16 - 16 \cdot \frac{\sqrt{3}}{2}\)\)
  \(c^2 = 20 - 8\sqrt{3}\)
  \(c = \sqrt{20 - 8\sqrt{3}}\) gives a single value of \( AC \).

Conclusion: Each statement individually gives us enough information to determine the length of \( AC \). Therefore, the correct answer is: Either statement (1) alone or statement (2) alone is sufficient to answer the question.