Question

unit 6 lesson 6

1. draw all of the diagonals in each regular polygon. fill in the table and find a pattern. is the pattern linear, exponential, or neither? how do you know? attempt to find an expression for the number of diagonals in an n-sided polygon.

table with columns: number of sides (3, 4, 5, 6, 7, 8, n) and number of diagonals
images of a triangle, square, pentagon, hexagon, heptagon, octagon

Explanation:

Step1: Analyze triangle (3 sides)

A triangle has no diagonals (a diagonal connects non - adjacent vertices; in a triangle, all vertices are adjacent). So for \(n = 3\), number of diagonals \(d=0\).

Step2: Analyze quadrilateral (4 sides)

A quadrilateral (e.g., square) has 2 diagonals. Using the formula logic: from each vertex, we can't connect to itself or its two adjacent vertices. So from each vertex, we can connect to \(4 - 3=1\) non - adjacent vertex. But if we do \(4\times(4 - 3)\), we double - count (since diagonal from A to B is same as B to A). So actual number is \(\frac{4\times(4 - 3)}{2}=2\).

Step3: Analyze pentagon (5 sides)

From each vertex, we can't connect to itself and 2 adjacent vertices. So from each vertex, we can connect to \(5 - 3 = 2\) non - adjacent vertices. Total non - double - counted diagonals: \(\frac{5\times(5 - 3)}{2}=\frac{5\times2}{2}=5\).

Step4: Analyze hexagon (6 sides)

Using the same logic: \(\frac{6\times(6 - 3)}{2}=\frac{6\times3}{2}=9\).

Step5: Analyze heptagon (7 sides)

\(\frac{7\times(7 - 3)}{2}=\frac{7\times4}{2}=14\).

Step6: Analyze octagon (8 sides)

\(\frac{8\times(8 - 3)}{2}=\frac{8\times5}{2}=20\).

Step7: General formula for \(n\) - sided polygon

For an \(n\) - sided polygon, from each of the \(n\) vertices, we exclude 3 vertices (itself and 2 adjacent vertices). So the number of diagonals from each vertex is \(n - 3\). But since each diagonal is counted twice (once from each end), the formula is \(d=\frac{n(n - 3)}{2}\).

Filling the table:

Number of Sides	Number of Diagonals
4	2
5	5
6	9
7	14
8	20
\(n\)	\(\frac{n(n - 3)}{2}\)

To check the pattern: The differences between consecutive number of diagonals: \(2-0 = 2\), \(5 - 2=3\), \(9 - 5 = 4\), \(14 - 9=5\), \(20 - 14 = 6\). The second differences (differences of differences) are constant ( \(3 - 2=1\), \(4 - 3 = 1\), etc.). So the pattern is quadratic (neither linear nor exponential) because the formula \(\frac{n(n - 3)}{2}=\frac{n^{2}-3n}{2}\) is a quadratic function.

Answer:

Number of Sides	Number of Diagonals
4	2
5	5
6	9
7	14
8	20
\(n\)	\(\boldsymbol{\frac{n(n - 3)}{2}}\)

The pattern is neither linear nor exponential (it is quadratic) because the formula for the number of diagonals is a quadratic function of \(n\) (\(\frac{n^{2}-3n}{2}\)) and the second differences of the number of diagonals are constant.