Find the total number of triangles in the given figure.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 37
The figure is a hexagon with its three longest diagonals and a triangle 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): ABC, ACF, ADE, AEF, BCG, CFG, DEI, EFI, FGH, FHI, GHJ, HIJ
Two parts (12): ABF, ABG, ADF, ADI, AFG, AFI, BFG, DFI, FGI, FGJ, FIJ, GIJ
Three parts (6): ACI, AEG, AGH, AHI, CGI, EGI
Four parts (6): ABI, ADG, AGJ, AIJ, BGI, DGI
Six parts (1): AGI
Total \(=12+12+6+6+1=37\) triangles.
Why the other options are incorrect
Option (1) 31 — Incorrect. This leaves out the 6 triangles made of four parts, such as BGI.
Option (3) 25 — Incorrect. This leaves out the 12 triangles made of two parts, such as ABG.
Option (4) 12 — Incorrect. This counts only the 12 smallest triangles, the ones that are a single part each.