Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $(QR)^2$ is equal to _______ -.

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] Take the lines $y=0$ and $y=5$ and $P(0,1)$. Let $Q=(q,0)$; rotating $Q$ about $P$ by $60^\circ$ gives $R$ with $y_R=1+\frac{\sqrt3}2q-\frac12=5$, so $q=3\sqrt3$. Then $(QR)^2=(PQ)^2=q^2+1=28$. Answer: 28.

Q. Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $(QR)^2$ is equal to _______ -.

Explanation

Reviewed by Dr. Abubacker

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