How many triangles are there in the given figure?

Concept

When several points on one straight line are joined to a common side vertex, triangles can be counted systematically by choosing pairs of points on that line.

Any two distinct points on the center line, together with a side vertex, form one triangle, provided the required straight segments are drawn.

Application

1. Label the five points on the vertical center line from top to bottom as P1, P2, P3, P4, and P5. Let the left and right vertices be L and R.

2. At L, every pair of center-line points forms one triangle. Thus the number is C(5, 2) = 10.

3. The same construction occurs at R, giving another C(5, 2) = 10 triangles.

4. For triangles spanning L and R, the horizontal segment LR is the base. P1, P2, P4, and P5 lie off this base, so they give 4 more triangles; P3 lies on LR and gives no triangle.

5. Therefore, the total is 10 + 10 + 4 = 24.

Cross-check

The 10 pairs of center-line points each produce one left triangle and one right triangle, giving 20. Adding the 4 triangles that span both side vertices again gives 24.

No further triangles are possible because all center points are collinear and every sloping segment terminates at L or R.

Therefore, the figure contains 24 triangles.