How many triangles are there in the following figure ?
12
The figure contains 12 triangles — option 1.
Step 1 — identify the smallest regions. The two cevians AD and BE meet at P, and PC is drawn from that point to the third vertex. Together they cut the big triangle into five smallest regions, each of which is itself a triangle :
APE, ABP, BPD, PDC, PEC — that is 5.
Step 2 — triangles made of two regions. Combine adjacent pairs and keep only those whose outline is a triangle :
| Combination | Triangle formed |
|---|---|
| ABP + APE | ABE — sides AB, BE, EA |
| ABP + BPD | ABD — sides AB, BD, DA |
| BPD + PDC | BPC — sides BP, PC, CB |
| APE + PEC | APC — sides AP, PC, CA |
That is 4 more. Note that PDC + PEC gives the quadrilateral PDCE, not a triangle, so it does not count.
Step 3 — triangles made of three regions.
| Combination | Triangle formed |
|---|---|
| APE + PEC + PDC | ADC — sides AD, DC, CA |
| BPD + PDC + PEC | BEC — sides BE, EC, CB |
That is 2 more.
Step 4 — the whole figure. All five regions together give ABC — 1 more.
Total :
\(5+4+2+1=12\)
The method to carry away. Count by the number of smallest regions a triangle contains — first those made of one region, then two, then three, and so on. Working in that order guarantees that nothing is counted twice and nothing is missed, and it is far safer than trying to spot triangles at random, which is how a candidate ends up at 10 or at 14.
Hence, the answer is 12.
Similar Questions
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Important Questions from Counting Figures
How many triangles are there in the given figure?
Find the number of squares in the given figure. ( Rectangles NOT to be included ).
Find the number of triangles in the given figure.
How many triangles are there in the given figure ?
How many triangles are there in the given figure?
