Find the total number of triangles in the given figure.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 30
The figure is a triangle with three 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 5 points, and any 2 of these points make a triangle with A.
Triangles on one line: \(\displaystyle\frac{5\times4}{2}=10\)
On BF (10): ABC, ABD, ABE, ABF, ACD, ACE, ACF, ADE, ADF, AEF
On GK (10): AGH, AGI, AGJ, AGK, AHI, AHJ, AHK, AIJ, AIK, AJK
On the base LP (10): ALM, ALN, ALO, ALP, AMN, AMO, AMP, ANO, ANP, AOP
Total \(=10\times3=30\) triangles.
Why the other options are incorrect
Option (1) 27 — Incorrect. This leaves out the 3 triangles whose base is made of four pieces.
Option (3) 18 — Incorrect. This leaves out the 12 triangles whose base is made of one piece.
Option (4) 10 — Incorrect. This counts only the 10 triangles standing on the base.