How many triangles are there in the figure given below?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 12
The figure is a square with one diagonal, both middle lines and a line joining the mid-points of two sides that meet at a corner. 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 (6): ABD, ADE, BDF, DEF, FGJ, FIJ
Two parts (4): ABE, ABF, AEF, BEF
Four parts (2): ACJ, AHJ
Total \(=6+4+2=12\) triangles.
Why the other options are incorrect
Option (2) 6 — Incorrect. This counts only the 6 smallest triangles, the ones that are a single part each.
Option (3) 8 — Incorrect. This leaves out the 4 triangles made of two parts, such as ABE.
Option (4) 10 — Incorrect. This leaves out the 2 triangles made of four parts, such as ACJ.