Question: ABC is an isosceles triangle with AB = AC = 10 units. If the area of the triangle is 48 square units, then what is the length of the base BC? Statement I: The length of BC is an even integer. Statement II: The height of the triangle is greater than the length of half of the base.

A Question is given followed by two Statements I and II. Consider the Question and the Statements. Which one of the following is correct in respect of the above Question and the Statements?

  1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone ✓
  2. B. The Question can be answered by using either Statement alone
  3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
  4. D. The Question can be answered even without using any of the Statements

Correct Answer: A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

Explanation

If base is 2x, height h = \sqrt{10^2 - x^2}. Area x\sqrt{100-x^2} = 48. Solving x^4 - 100x^2 + 2304 = 0 gives x^2 = 64 or 36, so x=8 or 6. The base 2x can be 16 or 12. Both are even integers, so Statement I is insufficient. Statement II specifies h > x. For x=6, h=8 (valid). For x=8, h=6 (invalid). Hence Statement II uniquely yields BC = 12.

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