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Determine how many diagonals each of the following polygons has. a. Nonagon b. Pentagon c. 17-gon d. n-gon

Determine how many diagonals each of the following polygons has. a. Nonagon b. Pentagon c. 17-gon d. n-gon

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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N-gone has how may diagonal
### Determining the Number of Diagonals in Polygons For each polygon, the number of diagonals can be determined using the formula: \[ D = \frac{n(n-3)}{2} \] Where \( D \) is the number of diagonals and \( n \) is the number of sides. #### Examples: a. **Nonagon** - A nonagon has 27 diagonals. b. **Pentagon** - A pentagon has 5 diagonals. c. **17-gon** - A 17-gon has 119 diagonals. d. **n-gon** - An n-gon has \(\frac{n(n-3)}{2}\) diagonals. (Type an expression using \( n \) as the variable.) This formula helps calculate the number of diagonals for any polygon based on its number of sides. ### Explanation of the Formula The formula \(\frac{n(n-3)}{2}\) comes from considering each vertex of the polygon, which can connect to every other vertex except itself and its two adjacent vertices (this is why we subtract 3). However, since we count each diagonal twice (once from each endpoint), we divide by 2 to get the correct number of distinct diagonals.
Transcribed Image Text:### Determining the Number of Diagonals in Polygons For each polygon, the number of diagonals can be determined using the formula: \[ D = \frac{n(n-3)}{2} \] Where \( D \) is the number of diagonals and \( n \) is the number of sides. #### Examples: a. **Nonagon** - A nonagon has 27 diagonals. b. **Pentagon** - A pentagon has 5 diagonals. c. **17-gon** - A 17-gon has 119 diagonals. d. **n-gon** - An n-gon has \(\frac{n(n-3)}{2}\) diagonals. (Type an expression using \( n \) as the variable.) This formula helps calculate the number of diagonals for any polygon based on its number of sides. ### Explanation of the Formula The formula \(\frac{n(n-3)}{2}\) comes from considering each vertex of the polygon, which can connect to every other vertex except itself and its two adjacent vertices (this is why we subtract 3). However, since we count each diagonal twice (once from each endpoint), we divide by 2 to get the correct number of distinct diagonals.
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