Counting the diagonals of a polygon
Counting the diagonals of a polygon
IP 1
7 marks
A diagonal of a polygon joins two vertices that are not next to each other.
- Explain why a convex polygon with \(n\) sides has \(\dfrac{n(n-3)}{2}\) diagonals.[3]
- A convex polygon has \(20\) diagonals. Find the number of sides and the sum of its interior angles.[2]
- Explain why no polygon has exactly \(30\) diagonals.[2]
Solution:
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Answer:(a) \(n-3\) diagonals from each of \(n\) vertices, each counted twice (b) \(8\) sides, \(1080^\circ\) (c) \(n(n-3)=60\) has no whole-number solution (\(54\) at \(n=9\), \(70\) at \(n=10\))
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