Find the total number of triangles in the given figure.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 44
The figure is a square with both diagonals, both middle lines 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 (16): ABD, ADF, BCE, BDG, BEG, CEH, DFG, EGH, FGI, FIK, GHJ, GIL, GJL, HJM, IKL, JLM
Two parts (16): ABF, ABG, AFG, BCG, BCH, BFG, BGH, CGH, FGK, FGL, FKL, GHL, GHM, GKL, GLM, HLM
Four parts (8): ACG, AGK, BFH, BFL, BHL, CGM, FHL, GKM
Eight parts (4): ACK, ACM, AKM, CKM
Total \(=16+16+8+4=44\) triangles.
Why the other options are incorrect
Option (1) 28 — Incorrect. This leaves out the 16 triangles made of two parts, such as ABF.
Option (2) 36 — Incorrect. This leaves out the 8 triangles made of four parts, such as ACG.
Option (3) 32 — Incorrect. This counts only the triangles made of one and of two parts.