How many triangles are there in the given figure?

Concept

To count every triangle in a compound figure, first count the smallest, undivided triangular regions, then add every larger triangle that is built from two or more of those regions but still has three complete straight sides that are actually drawn in the figure -- never a side that is only partly drawn or imagined. The total is simply (smallest triangles) + (every valid combination of them).

Application

1. The figure is two equal squares placed side by side, with a triangular tip attached to the outer side of the left square and another triangular tip attached to the outer side of the right square. One long horizontal line runs through the whole figure at mid-height. Inside each square, the mid-point of its outer vertical side is joined by two straight lines to the two corners on the far (inner) side of that square.

2. Left tip: the horizontal line splits this outer triangle into two smaller triangles, and the outer triangle itself is also a valid triangle -- 3 triangles in total from the left tip.

3. Right tip: by the same construction, the right tip also gives 3 triangles.

4. Left square: the two lines from the mid-point of its outer side to the two far corners create two corner triangles (one using the top edge, one using the bottom edge). Those same two lines, together with the vertical line that separates the two squares, enclose a third triangle in the middle of the square, which the vertical line's mid-point splits into two smaller triangles -- so the middle triangle contributes both its two halves and itself, undivided. Left square total: 2 (corner triangles) + 2 (middle-triangle halves) + 1 (undivided middle triangle) = 5.

5. Right square: the identical construction gives 5 triangles by the same reasoning.

6. Two further triangles use the full horizontal line as their base and stretch across the shared vertical edge between the two squares -- one with its apex where the squares' top edges meet, the other with its apex where their bottom edges meet. Their other two sides are exactly the diagonal lines already drawn inside each square, so both are genuine triangles, easy to miss if each square is checked in isolation.

7. Adding every category: 3 (left tip) + 3 (right tip) + 5 (left square) + 5 (right square) + 2 (spanning the middle) = 18.

Cross-check

Grouping the same triangles by size instead of by location gives an independent check: 12 undivided smallest triangles (2 in each tip, 4 in each square), 4 triangles built from exactly two of those smaller pieces (the whole tip in each of the two tips, the whole middle triangle in each of the two squares), and 2 triangles built from pieces spanning both squares. 12 + 4 + 2 = 18 -- the same total, confirming the count.