Let the distance between two parallel lines be 5 units and a point \( P \) lies between the lines at a unit distance from one of them. An equilateral triangle \( POR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _____________.
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Correct Answer: 48
Solution and Explanation
Step 1: Construct the equilateral triangle. Let the two parallel lines be \( y = 0 \) and \( y = 5 \), with \( P(0,1) \) lying between them. Since \( POR \) is an equilateral triangle, we use rotational symmetry to compute the coordinates of \( Q \) and \( R \).
Step 2: Compute the side length. Using coordinate transformations, we find the side length of \( \triangle POR \) is \( 4\sqrt{3} \).
Step 3: Compute \( QR^2 \). Since \( QR = 4\sqrt{3} \), squaring it gives: \[ (QR)^2 = 48. \]
Thus, the answer is \( \boxed{48} \).
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