How many triangles does the figure below contain?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 13
The figure is a triangle cut into nine 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 (9): ABC, BCE, BDE, CEF, DEH, DGH, EFI, EHI, FIJ
Four parts (3): ADF, BGI, CHJ
Nine parts (1): AGJ
Total \(=9+3+1=13\) triangles.
Why the other options are incorrect
Option (1) 10 — Incorrect. This leaves out the 3 triangles made of four parts, such as BGI.
Option (2) 12 — Incorrect. This leaves out the whole triangle AGJ, which is made of nine parts.
Option (4) 9 — Incorrect. This counts only the 9 smallest triangles, the ones that are a single part each.