How many diagonals can be drawn in total?
A normal-level "Count" problem.Count from one corner, multiply by the number of corners, then halve. The halving is because each line was counted from both of its ends.
GivenA 10-sided polygon (10 corners)
Hints
- Counting them all at once always misses some. Start with just one corner, A, and count how many lines come out of it.
- Here are lines from A to all 9 other corners. But some of them are not diagonals.
- The lines to the two neighbours are sides, not diagonals. Excluding those two and A itself, the diagonals from A number 10 − 3 = 7.
- The same holds at every corner. With 10 corners that gives 10 × 7 = 70 — but this count has a trap in it.
- Look at the thick line. It is counted among the 7 from A, and again among the 7 from F. Every diagonal gets counted twice, once from each end.
- So divide by two: 10 × 7 ÷ 2 = 35. The same formula works for any number of corners.
Answer 35
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 triangles are there in the figure?
- How many squares are there altogether?