Solution Guide for Chapter 12

Transcription

1 Solutio Guide for Chapter 1 Here are the solutios for the Doig the Math exercises i Girls Get Curves! DTM from p Here s a example we just draw 5 poits as vertices, draw (o-itersectig) lies betwee them like this, ad put dotted lies i for all the diagoals, ad we ca see how yep, 3 triagles are still formed! (Drawigs will vary.) 3. If a polygo has 10 sides, the from oe vertex, we ca draw diagoals to all vertices except itself ad its two eighbors, so that s 7 diagoals, ad that would create 8 triagles (it s always oe more triagle tha diagoal, whe drawig diagoals from a sigle vertex try for yourself!). 1

2 Ad how may total diagoals ca be draw i a 10-go from all the vertices? That s our formula: (! 3) à 10(10! 3) = 10(7) = 5(7) = 35 Aswer: 7; 8; Total diagoals draw i a 1000-go would be: (! 3) à 1000(1000! 3) = 1000(997) = Aswer: 498, If oe iterior agle of a regular polygo is 135, we ca use this formula to fid the umber of sides o the polygo: each it. agle = 180(! ), ad solve for! 135 = 180(! ) à 135 = 180( ) (we multiplied both sides by, ad the the deomiator cacels) à 135 = à 45 = 360 (we subtracted 180 from both sides) à =!360!45 = 8 So the polygo has 8 sides! For the secod part of this problem, we eed to fid the sum of the iterior agles. We could use the formula but sice we already kow each agle is 135 ad there are 8 of them, we ca just do: 135! 8 = 1080 Aswer: 8 sides; 1,080

3 6. If oe exterior agle of a regular polygo is 1, sice we kow the sum of all exterior agles of every polygo is always 360, the we ca just divide 360 by 1 ad get the umber of sides! = 30. So it has 30 sides. Now, what is the sum of the iterior agles? Well, if each exterior agle is 1, the each iterior agle must be = 168, right? Ad if there are 30 of them, the the sum of the iterior agles must be: 168! 30 = 5040 Aswer: 30 sides; 5, The total umber of degrees i a 11-go is give by the formula: 180( ), where = 11, so that s 180(11 ) = 180(9) = 160. I a regular 11-go, we ca fid the umber of degrees i each agle by just dividig by 11, ad we get: = or i fractio form, Aswer: 1,60 ; If the sum of the iterior agles i a polygo is 8,640, the we ca use this formula (ad solve for ) to fid the umber of sides is has: Sum of it. agles = 180 ( ). Let s do it! 8640 = 180 ( ) à = à 48 = à = 50 3

4 So it has 50 sides! (Note that it does t have to be a regular 50-go ay 50-go will have a sum of its exterior agles equal Picture the ( ) triagles we draw iside that will help you remember why this is true!) Aswer: 50 sides 9. The sum of exterior agles for ANY polygo is always 360. J Aswer: The formula for the umber of diagoals for a -go is: total # of diagoals = (! 3) solvig for : So if a polygo has a total 14 diagoals, we ca fid the umber of sides like this, 14 = (! 3) à 8 = ( 3) à 8 = 3 à 3 8 = 0 (We subtracted 8 from both sides ad the flipped the equatio) à ( + 4)( 7) = 0 (See Ch. 6 i Hot X: Algebra Exposed for factorig equatios!) à = 4, 7 Sice ca t be egative, the aswer is 7. We ve leared if a polygo has a total of 14 diagoals, it must have 7 sides. I other words, it s a heptago or 7-go. 4

5 Istead, if a polygo has a total of 0 diagoals, we d do the same exact thig with 0 istead of 14: 0 = (! 3) à 40 = ( 3) à 40 = 3 à 3 40 = 0 à ( + 5)( 8) = 0 à = 5, 8 Sice ca t be egative, the aswer is 8. We ve leared if a polygo has a total of 0 diagoals, it must have 8 sides. I other words, it s a octago or 8-go. Ad if a polygo has a total 35 diagoals, we ca fid the umber of sides like this, solvig for : 35 = (! 3) à 70 = ( 3) à 70 = 3 à 3 70 = 0 à ( + 7)( 10) = 0 à = 7, 10 Sice ca t be egative, the aswer is 10. We ve leared if a polygo has a total of 35 diagoals, it must have 10 sides. I other words, it s a decago or 10-go. Phew, doe! Aswer: heptago (7-go); octago (8-go); decago (10-go) 5

6 11. I a regular -go, sice the measure of each sigle exterior agle is 360, ad sice it must be supplemetary to each iterior agle, the we ca express each iterior agle of a regular -go like this: For part b, we ll simplify this aswer ito a sigle fractio by usig a commo deomiator, ad we ll do that by multiplyig the 180 by the copycat fractio (we assume does t equal 0, of course!), ad simply subtractig across the top. We get: = (180 )! 360 = 180! 360 = 180! 360 This is totally equal to what we started with; it s just writte differetly! Ad ow we ca simplify the umerator by factorig out the commo factor, 180, ad we get: 180(! ) Ad hey, that s the same formula as o p. 08 for the measure of a sigle iterior agle i a -go. Nice. (Check out Ch. 3 i Hot X: Algebra Exposed to review factorig with variables ad pullig out of ucool parties) Aswer: 11a b. 180( ) 6

7 1. If there are 0 people, ad everybody hugs each other, how may hugs are there? We ca imagie a 0-go, where a perso stads at each vertex (like o p.09, but with 0 sides), ad the each diagoal represets a hug. We ca t just blidly use the formula for diagoals, though, because with people, we ca hug our eighbors (ulike diagoals, which ever coect two eighbors). So for each perso, they ca hug 19 people, right? That s 19 diagoals we draw from that oe perso. We ca do that 0 times, which is 19(0) diagoals, except that we ve double couted, because me huggig you is the same hug as you huggig me! So we eed to divide this by, which is: 19(0) = 190. That s a lot of huggig. J Aswer: 190 hugs 7