Figure Figure 40.2

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1 40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called a cocave figure. Figure 40.1 shows a set of covex figures. Figure 40.1 O the other had, Figure 40.2 shows cocave figures. To show that a figure is cocave, it is eough to fid two poits withi the figure whose correspodig lie segmet is ot completely iside the figure. Figure 40.2 Regular Polygos By a closed curve we mea a curve that starts from oe poit ad eds i that same poit. By a simple curve we mea a curve that does ot cross itself. By a simple closed curve we mea a a curve i the plae that starts 1

2 ad eds i the same locatio without crossig itself. Several examples of curves are show i Figure Figure 40.3 A polygo is a simple closed curve made up of lie segmets. A polygo whose lie segmets are cogruet ad whose iterior agles are all cogruet is called a regular polygo. If a regular polygo cosists of sides the we will refer to it as regular -go. Figure 40.4 shows several tpyes of regular -gos. Figure

3 Agles of iterest i a regular -go are the followig: A vertex agle (also called a iterior agle) is formed by two cosecutive sides. A cetral agle is formed by the segmets coectig two cosecutive vertices to the ceter of the regular -go. A exterior agle is formed by oe side together with the extesio of a adjacet side as show i Figure 40.5 Figure 40.5 Agles Measures i Regular Polygos Let s first fid the measure of a cetral agle i a regular -go. Coectig the ceter of the -go to the vertices we create cogruet cetral agles. Sice the sum of the measures of the cetral agles is 360 the the measure of each cetral agle is 360. Next, we will fid the measure of each iterior agle of a regular -go. We will use the method of recogizig patters for that purpose. Sice the agles are cogruet the the measure of each is the sum of the agles divided by. Hece, we eed to fid the sum of the iterior agles. This ca be achieved by dividig the -go ito triagles ad usig the fact that the sum of the three iterior agles i a triagle is 180. The table below suggests a way for computig the measure of a vertex agle i a regular -go for =3,4,5,6,7,8. 3

4 So, i geeral, the measure of a iterior agle of a regular -go is ( 2) 180 = To measure the exterior agles i a regular -go, otice that the iterior agle ad the correspodig adjacet exterior agle are supplemetary. See Figure Thus, the measure of each exterior agle is 180 ( 2) 180 = 360. Figure 40.6 Example 40.1 (a) Fid the measure of each iterior agle of a regular decago (i.e., =10). (b) Fid the umber of sides of a regular polygo, each of whose iterior agles has a measure of

5 Solutio. (a) The measure of each agle is: (10 2) 180 (b) We are give that ( 2) 180 that 360 = = 175 or = 72 5 = = 5. Thus, = 360 Practice Problems = 175. This implies Problem 40.1 List the umerical values of the shapes that are covex. Problem 40.2 Determie how may diagoals each of the followig has: (a) 20-go (b) 100-go (c) -go Problem 40.3 I a regular polygo, the measure of each iterior agle is 162. How may sides does the polygo have? Problem 40.4 Two sides of a regular octago are exteded as show i the followig figure. Fid the measure of 1. Problem 40.5 Draw a quadrilateral that is ot covex. Problem 40.6 What is the sum of the iterior agle measures of a 40-go? 5

6 Problem 40.7 A Caadia ickel has the shape of a regular dodecago (12 sides). How may degrees are i each iterior agle? Problem 40.8 Is a rectagle a regular polygo? Why or why ot? Problem 40.9 Fid the measures of the iterior, exterior, ad cetral agles of a 12-go. Problem Suppose that the measure of the iterior agle of a regular polygo is 176. What is the measure of the cetral agle? Problem The measure of the exterior agle of a regular polygo is 10. How may sides does this polygo have? Problem The measure of the cetral agle of a regular polygo is 12. How may sides does this polygo have? Problem The sum of the measures of the iterior agles of a regular polygo is How may sides does the polygo have? Problem How may lies of symmetry does each of the followig have? (a) a regular petago (b) a regular octago (c) a regular hexago. Problem How may rotatioal symmetry does a petago have? 6

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