How many triangles does the figure below contain?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 20
The figure is a square with both diagonals and a square joining the mid-points of 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 (8): ABD, ADF, BCE, CEH, FIK, HJM, IKL, JLM
Two parts (4): ABF, BCH, FKL, HLM
Three parts (4): ACG, AGK, CGM, GKM
Six parts (4): ACK, ACM, AKM, CKM
Total \(=8+4+4+4=20\) triangles.
Why the other options are incorrect
Option (1) 8 — Incorrect. This counts only the 8 smallest triangles, the ones that are a single part each.
Option (3) 12 — Incorrect. This counts only the triangles made of one and of two parts.
Option (4) 16 — Incorrect. This leaves out the 4 triangles made of three parts, such as ACG.