How many triangles are there in the figure given below?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 15
The figure is a triangle with one line from the top corner to the base and two lines from the left corner to the right 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 (4): ABC, ACF, CEF, EFG
Two parts (6): ABF, ADE, AEF, BDF, CFG, DFH
Three parts (2): AFG, AGH
Four parts (2): ADF, BFH
Six parts (1): AFH
Total \(=4+6+2+2+1=15\) triangles.
Why the other options are incorrect
Option (2) 14 — Incorrect. This leaves out the whole triangle AFH, which is made of six parts.
Option (3) 13 — Incorrect. This leaves out the 2 triangles made of four parts, such as ADF.
Option (4) 12 — Incorrect. This leaves out both the triangles made of six parts and those made of four parts.