Count the number of triangles in the figure below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 32
The figure is a hexagon with the six-pointed star made by joining every second corner. 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 (12): ABE, ACD, ADE, BEG, BGK, CDF, CFH, FHI, GJK, HIL, IJL, JKL
Two parts (12): ABD, ABG, ACE, ACF, BEK, BJK, CDH, CHI, FHL, GKL, HJL, IKL
Three parts (6): ABC, ABK, ACH, BKL, CHL, HKL
Four parts (2): AHK, BCL
Total \(=12+12+6+2=32\) triangles.
Why the other options are incorrect
Option (1) 26 — Incorrect. This leaves out the 6 triangles made of three parts, such as ACH.
Option (2) 12 — Incorrect. This counts only the 12 smallest triangles, the ones that are a single part each.
Option (3) 20 — Incorrect. This leaves out the 12 triangles made of two parts, such as ACF.