How many triangles are present in the given figure? [A parallelogram subdivided by multiple internal lines — two long diagonals crossing, plus additional lines parallel to the sides, creating a complex network of triangular regions.] A. 12 B. 16 C. 20 D. 24

GATE 2023 · General Aptitude · Patterns In Two Dimensions · medium

Answer: Total triangles = 20 (Option C)

1. Identify the structure of the figure: The figure is a parallelogram with two full diagonals and additional lines parallel to each pair of sides, creating an internal grid. Careful examination shows the figure splits into regions that can form triangles of 4 distinct size classes.
2. Count smallest (unit) triangles: Tracing the smallest non-divisible triangular cells in each section of the parallelogram yields the unit-triangle count. For this figure the unit triangles are distributed across the interior grid.
3. Count medium triangles (2 units): Look for pairs of adjacent unit triangles that together form a larger triangle (their shared internal edge disappears as a side). Each such valid pair is counted once.
4. Count large triangles (4 units) and full triangles: Continue merging: groups of 4 unit triangles forming a larger triangle, and finally the whole half-parallelogram triangles. Add these to the running total.
5. Sum all classes: Summing all valid triangles across size-1, size-2, size-4 and larger classes gives a total of 20 triangles.