GATE EE 2022 Set 1 — Question 9

Label the vertices of the regular hexagon $V_1, V_2, \dots, V_6$ in order, with$$V_1=(1,0),\ V_2=(0.5,\tfrac{\sqrt3}{2}),\ V_3=(-0.5,\tfrac{\sqrt3}{2}),\ V_4=(-1,0),\ V_5=(-0.5,-\tfrac{\sqrt3}{2}),\ V_6=(0.5,-\tfrac{\sqrt3}{2}).$$Only three segment lengths occur: sides ($a$), short diagonals ($\sqrt3\,a$) and long diagonals ($2a$).Step 1 - use statements 1 and 2.
Two sides, two short diagonals or two long diagonals are never perpendicular to each other (sides and long diagonals run in the directions $0^\circ, 60^\circ, 120^\circ$, short diagonals in $30^\circ, 90^\circ, 150^\circ$). A side and a short diagonal can be perpendicular, but only when they share a vertex (e.g. $V_2V_3 \perp V_2V_6$). That is impossible here, because $R, S, P, Q$ are four different vertices. So the only possibility is that $RS$ is a long diagonal and $PQ$ is a short diagonal perpendicular to it with no common vertex. For example, $V_1V_4$ is perpendicular to both $V_2V_6$ and $V_3V_5$.By the symmetry of the hexagon we may take $R = V_4,\ S = V_1$ and $\{P,Q\} = \{V_2,V_6\}$, so $\{T,U\} = \{V_3,V_5\}$. (The choice $\{P,Q\} = \{V_3,V_5\}$ is the mirror image of this one in the vertical line through the centre, with $R$ and $S$ interchanged, and leads to the same conclusions.)Step 2 - use statement 3 ($RU \parallel TQ$).
With $U = V_3$, $RU = V_4V_3$ is a side with direction $(0.5,\tfrac{\sqrt3}{2})$. Among the remaining choices only $TQ = V_5V_2$ (a long diagonal, direction $(1,\sqrt3) \propto (0.5,\tfrac{\sqrt3}{2})$) is parallel to it. Hence$$P=V_6,\quad Q=V_2,\quad R=V_4,\quad S=V_1,\quad T=V_5,\quad U=V_3 .$$(The mirror assignment $U=V_5,\ T=V_3,\ Q=V_6,\ P=V_2$ is the only other possibility and gives the same conclusions.)Step 3 - test the options.

• (A) $RT = V_4V_5$ has direction $(0.5,-\tfrac{\sqrt3}{2})$ and $QS = V_2V_1$ has direction $(0.5,-\tfrac{\sqrt3}{2})$. They are opposite sides of the hexagon, hence parallel - correct.
• (B) $TQ = V_5V_2 \propto (0.5,\tfrac{\sqrt3}{2})$ while $PU = V_6V_3 \propto (-0.5,\tfrac{\sqrt3}{2})$; not parallel.
• (C) $RP = V_4V_6 \propto (\tfrac{\sqrt3}{2},-0.5)$ and $UQ = V_3V_2 = (1,0)$; their dot product is $\tfrac{\sqrt3}{2} \neq 0$, so not perpendicular.
• (D) $QS \propto (0.5,-\tfrac{\sqrt3}{2})$ and $RP \propto (\tfrac{\sqrt3}{2},-0.5)$; dot product $= 0.433+0.433 = 0.866 \neq 0$, so not perpendicular.

Only statement (A) follows from the given conditions.Answer: (A) The line segment joining R and T is parallel to the line segment joining Q and S.