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

The correct answer is option (D): Either statement (1) alone or statement (2) alone is sufficient to answer the question

Let's analyze the problem and the statements provided. We are given a triangle ABC with AB = 2 cm and BC = 4 cm. We need to determine the length of AC.

Statement 1: The three sides are in geometric progression (GP).

If the sides are in GP, we consider possible orders of the three sides (since we only know AB and BC):

1. Case 1: 2, AC, 4.
   Then AC/2 = 4/AC ⇒ AC² = 8 ⇒ AC = 2√2.
2. Case 2: AC, 2, 4.
   Then 2/AC = 4/2 ⇒ AC = 1 (reject — triangle inequality fails).
3. Case 3: 2, 4, AC.
   Then 4/2 = AC/4 ⇒ AC = 8 (reject — triangle inequality fails).

Check triangle inequality for the valid candidate AC = 2√2:

• 2 + 2√2 > 4 — true
• 2 + 4 > 2√2 — true
• 4 + 2√2 > 2 — true

Thus Statement 1 alone yields a unique valid value AC = 2√2, so Statement 1 is sufficient.

Statement 2: ∠ABC = 30°.

Use the Law of Cosines on triangle ABC:
AC² = AB² + BC² − 2·AB·BC·cos(∠ABC).

Substitute AB = 2, BC = 4, ∠ABC = 30°:

AC² = 2² + 4² − 2·2·4·cos 30°
    = 4 + 16 − 16·(√3/2)
    = 20 − 8√3
AC = √(20 − 8√3)
  

The expression √(20 − 8√3) is a definite numerical value (approximately 2.48). Therefore Statement 2 also determines AC uniquely, so Statement 2 alone is sufficient.

Conclusion

Each statement by itself suffices to determine AC. Hence the correct option is:
(D) Either statement (1) alone or statement (2) alone is sufficient.