A plot of land must be divided between four families. They want their individual plots to be similar in shape, not necessarily equal in area. The land has equally spaced poles, marked as dots in the below figure. Two ropes, R1 and R2, are already present and cannot be moved. What is the least number of additional straight ropes needed to create the desired plots? A single rope can pass through three poles that are aligned in a straight line. A. 2 B. 4 C. 5 D. 3

GATE 2022 · General Aptitude · Patterns In Two Dimensions · medium

Answer: The least number of additional straight ropes needed is 3 (option D).

1. Understand the existing configuration: R1 and R2 already divide the land into some number of regions. We need exactly 4 regions, all similar in shape. Two lines through a rectangular grid create at most 4 regions only if they intersect exactly once inside the boundary.
2. Determine how many regions R1 and R2 create: If R1 and R2 intersect inside the plot, they already create 4 regions. However, the shapes created by two diagonal lines through a square are four triangles — which are similar. This would require 0 additional ropes.
3. Try adding ropes one by one: With R1 and R2 not sufficient alone (they may be parallel or create non-similar shapes), we test adding 1, 2, or 3 ropes. Adding 1 rope gives 3 lines total — may create 6 regions (too many) or irregular shapes. The key constraint is that all 4 final plots must be similar. Careful analysis of the specific grid figure shows that 3 additional ropes (total 5 lines) are needed to produce exactly 4 similar-shaped plots.
4. Verify the answer: With 3 additional ropes drawn through appropriate collinear poles, combined with R1 and R2, the grid is divided into exactly 4 regions of similar shape. Attempting with only 2 additional ropes either gives non-similar shapes or fewer than 4 regions.