Count the number of triangles in the figure below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 33
The figure is a rectangle made of two squares with the two long diagonals of the rectangle and one diagonal of each square. 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 (10): ABD, ADE, AEG, BCD, CDF, CFI, DEH, DFH, EGH, FHI
Two parts (9): ACD, ADG, ADH, AGH, CDH, CDI, CHI, DGH, DHI
Three parts (6): ABH, ACE, ACF, AFH, BCH, CEH
Four parts (5): ACG, ACI, AHI, CGH, DGI
Six parts (3): ACH, AGI, CGI
Total \(=10+9+6+5+3=33\) triangles.
Why the other options are incorrect
Option (1) 19 — Incorrect. This counts only the triangles made of one and of two parts.
Option (2) 24 — Incorrect. This leaves out the 9 triangles made of two parts, such as ACD.
Option (3) 28 — Incorrect. This leaves out the 5 triangles made of four parts, such as ACI.