Question

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 _____________.

Hint:

For geometric problems involving equilateral triangles and parallel lines, use coordinate geometry and rotational transformations to simplify calculations.

Answer 1

Correct Answer: 48

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 \( \boxed{48} \).