How many triangles are there in the figure given below?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 35
The figure is a pentagon with all five of its diagonals. 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): ABC, ACD, ADE, BCF, BFI, DEG, EGJ, FHI, GHJ, HIJ
Two parts (10): ABD, ABF, ACE, AEG, BCI, BHI, DEJ, EHJ, FIJ, GIJ
Three parts (10): ABE, ABI, AEJ, AFJ, AGI, BDJ, BEH, BIJ, CEI, EIJ
Five parts (5): ABJ, AEI, AIJ, BEI, BEJ
Total \(=10+10+10+5=35\) triangles.
Why the other options are incorrect
Option (2) 30 — Incorrect. This leaves out the 5 triangles made of five parts, such as ABJ.
Option (3) 25 — Incorrect. This leaves out the 10 triangles made of three parts, such as ABI.
Option (4) 20 — Incorrect. This leaves out both the triangles made of five parts and those made of three parts.