Consider a square ABCD with midpoints E, F, G, H of AB, BC, CD and DA respectively. Let L denote the line passing through F and H. Consider points P and Q, on L and inside ABCD, such that the angles APD and BQC both equal 120°. What is the ratio of the area of ABQCDP to the remaining area inside ABCD?
Explanation:
Let the length of the sides of the square be 2s.
Consider ∆BQF,
BF = s
In 30° - 60° - 90° triangle,
QF = s3
∴ Area of ∆BQF = 12 × s × s3
Area of ABQCDP = Area of square ABCD – 4 × Area of ∆BQF =4s2-412×s×s3
∴ Required ratio = 4s2-412×s×s3412×s×s3 = 23-1
Hence, option (e).
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