How many triangles are there in the figure given below?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 27
The figure is a triangle with two lines from each base corner to the opposite side. 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 (5): EGJ, FHK, GIJ, HIK, IJK
Two parts (8): BDJ, CDK, DGK, DHJ, EIJ, FIK, GJK, HJK
Three parts (6): ABK, ACJ, BEK, CFJ, EJK, FJK
Four parts (3): BHJ, CGK, DJK
Six parts (4): AEK, AFJ, BJK, CJK
Nine parts (1): AJK
Total \(=5+8+6+3+4+1=27\) triangles.
Why the other options are incorrect
Option (1) 23 — Incorrect. This leaves out the 4 triangles made of six parts, such as AFJ.
Option (2) 21 — Incorrect. This leaves out the 6 triangles made of three parts, such as ACJ.
Option (3) 24 — Incorrect. This leaves out the 3 triangles made of four parts, such as DJK.