How many triangles are there in the figure?
A hard-level "Count" problem.To count shapes, do not chase them with your eyes — find out what has to be chosen to pin one down. Here it was two points on the base.
GivenLines are drawn from apex A to points on the base. There are 5 points on the base in all.
Hints
- It is not only the little triangles: the wider ones spanning several sections, and the whole outer triangle, all count too. Miss these and the total falls far short.
- Stop counting and ask what determines a triangle. Every one has A as its apex; the only thing that varies is which two base points are chosen.
- And conversely, choosing two base points fixes one triangle. The question has become "how many ways to choose 2 points out of 5".
- The first point has 5 choices and the second 4, giving 5 × 4 = 20. But "B then D" and "D then B" are the same triangle counted twice, so divide by 2: 10.
Answer 10
More "Count" problems
- How many turns does the small circle make?
- How many cubes have 2 painted faces?
- Into how many pieces do 4 lines cut the paper?
- In how many ways can the three chosen rods form a triangle?
- How many squares are there altogether?
- How many squares does the diagonal pass through?