Question

set

1. draw the lines of symmetry for each regular polygon. fill in the table, including an expression for the number of lines of symmetry in an n - sided polygon.

number of sides	number of lines of symmetry
4
5
6
7
8
n

polygons: triangle, square, pentagon, hexagon, heptagon, octagon

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.

number of sides	number of diagonals
4
5
6
7
8
n

polygons: triangle, square, pentagon, hexagon, heptagon, octagon

Answer:

Problem 4: Lines of Symmetry in Regular Polygons

Step 1: Recall the property of regular polygons

A regular \( n \)-sided polygon has \( n \) lines of symmetry. This is because a regular polygon is symmetric about lines that pass through the center and either a vertex and the midpoint of the opposite side (for even \( n \)) or through two opposite vertices (for odd \( n \)), and there are \( n \) such lines.

Step 2: Fill the table for specific values

• For \( n = 3 \) (equilateral triangle), number of lines of symmetry is \( 3 \).
• For \( n = 4 \) (square), number of lines of symmetry is \( 4 \).
• For \( n = 5 \) (regular pentagon), number of lines of symmetry is \( 5 \).
• For \( n = 6 \) (regular hexagon), number of lines of symmetry is \( 6 \).
• For \( n = 7 \) (regular heptagon), number of lines of symmetry is \( 7 \).
• For \( n = 8 \) (regular octagon), number of lines of symmetry is \( 8 \).
• For general \( n \), number of lines of symmetry is \( n \).

Problem 5: Diagonals in Regular Polygons

Step 1: Recall the formula for diagonals

The number of diagonals \( d \) in an \( n \)-sided polygon is given by the formula \( d=\frac{n(n - 3)}{2} \). We can derive this as follows: each vertex connects to \( n-3 \) other vertices (excluding itself and its two adjacent vertices), and since each diagonal is counted twice (once from each end), we divide by 2.

Step 2: Calculate diagonals for specific \( n \)

• For \( n = 3 \) (triangle): \( d=\frac{3(3 - 3)}{2}=\frac{3\times0}{2} = 0 \)
• For \( n = 4 \) (quadrilateral): \( d=\frac{4(4 - 3)}{2}=\frac{4\times1}{2}=2 \)
• For \( n = 5 \) (pentagon): \( d=\frac{5(5 - 3)}{2}=\frac{5\times2}{2}=5 \)
• For \( n = 6 \) (hexagon): \( d=\frac{6(6 - 3)}{2}=\frac{6\times3}{2}=9 \)
• For \( n = 7 \) (heptagon): \( d=\frac{7(7 - 3)}{2}=\frac{7\times4}{2}=14 \)
• For \( n = 8 \) (octagon): \( d=\frac{8(8 - 3)}{2}=\frac{8\times5}{2}=20 \)
• For general \( n \), \( d = \frac{n(n - 3)}{2} \)

Step 3: Determine the pattern type

To check if the pattern is linear or exponential, we look at the formula \( d=\frac{n(n - 3)}{2}=\frac{n^{2}-3n}{2} \), which is a quadratic function (degree 2). A linear function has degree 1, and an exponential function has the form \( a\times b^{n} \). Since the formula for diagonals is quadratic, the pattern is neither linear nor exponential (it is quadratic). We know this because the second difference of the number of diagonals is constant (for a quadratic sequence, the second difference is constant). Let's verify with the values:

• For \( n = 3,4,5,6,7,8 \), the number of diagonals are \( 0,2,5,9,14,20 \)
• First differences: \( 2 - 0=2 \), \( 5 - 2 = 3 \), \( 9 - 5=4 \), \( 14 - 9 = 5 \), \( 20 - 14=6 \)
• Second differences: \( 3 - 2=1 \), \( 4 - 3 = 1 \), \( 5 - 4=1 \), \( 6 - 5 = 1 \)

Since the second difference is constant (1), the sequence is quadratic (neither linear nor exponential).

Final Tables

Table for Lines of Symmetry

Number of Sides	Number of Lines of Symmetry
4	4
5	5
6	6
7	7
8	8
\( n \)	\( n \)

Table for Diagonals

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

Pattern for Diagonals

The pattern is neither linear nor exponential (it is quadratic). We know this because the second difference of the number of diagonals is constant (equal to 1), which is a characteristic of a quadratic sequence. The formula for the number of diagonals in an \( n \)-sided polygon is \( \boldsymbol{\frac{n(n - 3)}{2}} \).