Triangle with two lines from the top corner and one line across the middle.

Solution: 4 lines from the top, 2 lines across: 242 = 12.

What are the common traps in Count triangles / squares / rectangles questions?

Big ones missed: the triangle made of two or four pieces is still a triangle. False triangles: a region with a bent side is not a triangle. Double counting: list each triangle once by fixing an order of corners.

Start here · the first lesson of Count triangles / squares / rectangles

Count triangles / squares / rectangles

Non-verbal Reasoning · for IPMAT Indore, IPMAT Rohtak and JIPMAT

The idea

Count the triangles, squares or rectangles whose every side lies on a drawn line. JIPMAT asks one each year: triangles in 2024 (Q23, 21 triangles) and 2026 (Q17, 11 triangles), squares and rectangles in 2025 (Q2).

Textbook vs exam method

Textbook method

Letter every corner and every crossing point.
Take the corners in order (A, B, C, …). For each, list the triangles that have it as their first corner, using only later letters.
A triple is a triangle only if each pair of its corners lies on one drawn straight line and the three are not on one line.
Add the counts.

Exam method

Fan + cross lines. $k$ lines from one corner and $t$ lines crossing all of them give $t\binom{k}{2}$ triangles.
Grids. squares size by size ( $4 \times 4$ : $16 + 9 + 4 + 1 = 30$ ); rectangles $\binom{m+1}{2}\binom{n+1}{2}$ .
Square with both diagonals and both midlines. 8 + 4 + 4 = 16 triangles.
Anything else. split the figure into parts you know and count the triangles that cross between parts separately.