If p and q are real numbers between 0 and 1 such that the points (p,1), (1,q) and (0,0) form an equilateral triangle, then what is (p+q) equal to?

  1. A. \sqrt{2}
  2. B. \sqrt{2}-1
  3. C. 2-\sqrt{3}
  4. D. 4-2\sqrt{3} ✓

Correct Answer: D. 4-2\sqrt{3}

Explanation

Let O(0,0), A(p,1), B(1,q). OA^2 = OB^2 \implies p^2+1 = 1+q^2 \implies p=q (since both are in (0,1)). Also OA^2 = AB^2 \implies p^2+1 = (p-1)^2+(1-p)^2 \implies p^2+1 = 2p^2-4p+2 \implies p^2-4p+1=0. The roots are 2 \pm \sqrt{3}. Since p \in (0,1), p = 2-\sqrt{3}. Thus p+q = 2p = 4-2\sqrt{3}.

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