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(a) Use a straight edge to draw the regular stars made from these regular polygons. [Image_1: Polygon images for drawing stars]
(b) Draw the starting polygon inside this regular star. [Image_2: Regular star image]
(c) (i) Complete this table. [Table_1: Number of sides of polygons and stars]
\begin{align} \text{Number of sides (P) of the starting polygon} & & \text{Number of sides (S) of the star} \\ 5 & & 10 \\ 6 & & \\ 7 & & \\ 8 & & \\ 9 & & \end{align}
(ii) Write down a formula for $S$ in terms of $P$.
(d) (i) Complete the table. [Table_2: Regular star and point angles]
\begin{align} \text{Regular star} & & \text{Number of points} & & \text{Sum of star's point angles} \\ \star & & 5 & & 180^\circ \\ \star & & 6 & & 360^\circ \\ \star & & 7 & & 540^\circ \\ \star & & 8 & & 720^\circ \\ \star & & 9 & & \end{align}
(ii) Is it possible for a regular star, made from a regular polygon, to have the sum of its point angles equal to $1450^\circ$? Explain how you decide.
(e) (i) The regular pentagon making a regular star is shown in bold. The sum of the interior angles of a pentagon is $540^\circ$. Use this information to calculate the value of $p$. [Image_3: Diagram showing regular pentagon]
(ii) This diagram shows part of a different regular star. It also shows, in bold, part of the regular polygon that makes it. Find an equation connecting $a$ and $b$. Write your answer in its simplest form. [Image_4: Diagram showing angle a and b]
560
553
538
You can also make stars by placing two congruent regular polygons on top of each other and rotating one of the polygons about their common centre. For example and and
(a) Complete this table.
[Table_1]
(b) Write down an equation connecting $P$ and $S$.
595
541
502
You can also make regular stars by joining dots that are equally spaced round a circle. Here is a star made by joining every second dot round a circle with 5 equally spaced dots.
This 3-point star is made by connecting every second dot round a circle with 6 equally spaced dots.
Regular polygons are also regular stars and their vertices are the points of the star.
(a) Draw the stars made by connecting every second dot round these circles.
[Image_3: 7 dots]
[Image_4: 8 dots]
[Image_5: 9 dots]
[Image_6: 10 dots]
Complete this table.
[Table_1]
| Number of equally spaced dots | Number of points of the star |
|-----------------------------|-----------------------------|
| 5 | 5 |
| 6 | 3 |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| 11 | |
| 12 | |
(b) There are 370 equally spaced dots round a circle. Every second dot is joined. Find the number of points of the star.
525
577
519
In question 3 you made stars by joining every second dot round a circle. You can also make stars by joining every third dot. Starting from 1, dots are numbered clockwise. This gives a way to code the star. 1 \rightarrow 4 \rightarrow 7 \rightarrow 2 \rightarrow 5 \rightarrow 8 \rightarrow 3 \rightarrow 6 \rightarrow 1
(a) Here is the code for a star. 1 \rightarrow 4 \rightarrow 2 \rightarrow 5 \rightarrow 3 \rightarrow 1
(i) Draw this star on the diagram below.
(ii) Write down a different code for the star you have drawn.
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(b) Here are some more stars and their codes.
1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5 \rightarrow 6 \rightarrow 7 \rightarrow 8 \rightarrow 1
1 \rightarrow 3 \rightarrow 5 \rightarrow 7 \rightarrow 1
1 \rightarrow 4 \rightarrow 7 \rightarrow 2 \rightarrow 5 \rightarrow 8 \rightarrow 3 \rightarrow 6 \rightarrow 1
(i) Write down the connection between the code and the number of points of the star.
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(ii) Write down the connection between the code and the number of dots round the circle.
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(c) Make a sketch showing the numbered dots and the regular star with this code.
1 \rightarrow 4 \rightarrow 7 \rightarrow 1
(d) Find three codes, each starting with 1, which make a star with 10 points.
You may use these circles to help you.
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573
591
507
