How many triangles are present in the given figure?
Show Hint
Avoid double counting by (i) fixing a scale (unit, 2-unit, spanning) and (ii) counting panel-wise. Alternatively, enumerate by apex: list all triangles from each allowed vertex, then move to the next.
Step 1: Decompose into panels. Two nearly-vertical segments split the slanted outer quadrilateral into three slanted panels. Two oblique lines traverse all panels. Intersections of top, bottom with the two obliques and the two verticals create repeatable triangular cells.
Step 2: Count unit triangles (smallest). Each panel cut by the two obliques contains four unit triangles (two up, two down). Therefore \[ N_{\text{unit}}=3\times 4=12. \]
Step 3: Count size–2 triangles within a panel. In each panel, pairs of adjacent unit triangles along an oblique combine to form two larger triangles (one up, one down). Hence \[ N_{\text{size-2, within}}=3\times 2=6. \]
Step 4: Count size–2 triangles across panel boundaries. Across each of the two vertical boundaries, a unit triangle from the left panel can pair with its touching unit from the right panel (both orientations). Thus \[ N_{\text{size-2, across}}=2\times 2=4. \]
Step 5: Count the largest spanning triangles. Using full panel height with both obliques we obtain four additional distinct large triangles (two on the left half, two on the right half): \[ N_{\text{largest}}=4. \]
Step 6: Sum without double counting (disjoint constructions). \[ N_{\triangle}=12+6+4+4=\boxed{24}. \]
