Count the number of triangles in the figure below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 27
The figure is a triangle cut into sixteen small triangles by lines parallel to its sides. A triangle is counted when all three of its sides lie on lines of the figure. Naming the corners and crossing points and counting by the number of small parts each triangle covers makes sure none is missed or counted twice.
One part (16): ABC, BCE, BDE, CEF, DEH, DGH, EFI, EHI, FIJ, GHL, GKL, HIM, HLM, IJN, IMN, JNO
Four parts (7): ADF, BGI, CHJ, DFM, DKM, ELN, FMO
Nine parts (3): AGJ, BKN, CLO
Sixteen parts (1): AKO
Total \(=16+7+3+1=27\) triangles.
Why the other options are incorrect
Option (1) 23 — Incorrect. This leaves out both the triangles made of sixteen parts and those made of nine parts.
Option (2) 16 — Incorrect. This counts only the 16 smallest triangles, the ones that are a single part each.
Option (3) 20 — Incorrect. This leaves out the 7 triangles made of four parts, such as DKM.