How many triangles does the figure below contain?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 18
The figure is a triangle with two lines from the top corner and two lines across it. Every triangle in it has the top corner A as one corner and its base on one of the lines across, the base included. On each such line the lines from A meet 4 points, and any 2 of these points make a triangle with A.
Triangles on one line: \(\displaystyle\frac{4\times3}{2}=6\)
On BE (6): ABC, ABD, ABE, ACD, ACE, ADE
On FI (6): AFG, AFH, AFI, AGH, AGI, AHI
On the base JM (6): AJK, AJL, AJM, AKL, AKM, ALM
Total \(=6\times3=18\) triangles.
Why the other options are incorrect
Option (1) 9 — Incorrect. This counts only the 9 smallest triangles, whose base is a single piece.
Option (2) 6 — Incorrect. This counts only the 6 triangles standing on the base.
Option (3) 12 — Incorrect. This leaves out the 6 triangles whose base is made of two pieces.