How many triangles are there in the figure given below?

Concept — In any figure made of straight line segments, a triangle is a closed shape bounded by exactly three of the drawn segments; it may be one smallest undivided region or several adjacent regions merged together. The dependable way to count them is by size class: count the smallest undivided triangular regions first, then the triangles formed by merging two such regions, then three, and so on, and add the classes at the end. Working class by class guarantees that nothing is missed and nothing is counted twice.

Applying this to the figure — label the inverted outer triangle A (top-left), B (top-right) and C (bottom vertex). M is the mid-point of the top side AB. The horizontal segment joins D on side AC to E on side BC, and the vertical segment MC crosses it at F. The drawn lines are therefore AB (through M), AC (through D), BC (through E), DE (through F), MD, ME and MC (through F).

1. Smallest undivided regions: AMD, MBE, MDF, MFE, DFC and FEC. Each is bounded by three drawn segments and has no line running through it, so this class gives 6.

2. Triangles made of two adjacent regions: MDF + MFE = MDE, MDF + DFC = MDC, MFE + FEC = MEC, and DFC + FEC = DEC. This class gives 4.

3. Triangles made of three adjacent regions: AMD + MDF + DFC = AMC, and MBE + MFE + FEC = MBC. This class gives 2.

4. Triangle made of all six regions: the whole outer triangle ABC. This class gives 1.

5. Add the classes: 6 + 4 + 2 + 1 = 13.

Cross-check — sort the same triangles independently by whether they use the bottom vertex C.

• Using C: ABC, AMC, MBC, MDC, MEC, DEC, DFC, and FEC — 8 triangles.

• Not using C: AMD, MBE, MDF, MFE, and MDE — 5 triangles.

The independent classification gives 8 + 5 = 13, confirming the size-class total.

The figure therefore contains 13 triangles.