Chapter44. Polygons and solids. Contents: A Polygons B Triangles C Quadrilaterals D Solids E Constructing solids

Transcription

1 Chpter44 Polygons nd solids Contents: A Polygons B Tringles C Qudrilterls D Solids E Constructing solids

2 74 POLYGONS AND SOLIDS (Chpter 4) Opening prolem Things to think out: c Wht different shpes cn you see in this digrm? The tringles ech hve 2 sides of equl length. Wht nme do we give to tringles like this? If we fold the shpe long the dotted lines, wht type of solid cn we construct? In this chpter we will explore two-dimensionl polygons, nd three-dimensionl solids. A A polygon is closed figure which hs only stright line sides nd which does not cross itself. POLYGONS A closed figure hs no gps in it. Here re some polygons: Here re some exmples of polygons tht we often see round us: GIVE WAY A polygon must lie in flt surfce. We cll this surfce plne. These figures re not polygons: curved side crosses itself not closed

3 NAMING POLYGONS We nme polygons ccording to how mny sides nd ngles they hve: POLYGONS AND SOLIDS (Chpter 4) 75 tringle 3 sides qudrilterl 4 sides pentgon 5 sides hexgon 6 sides heptgon 7 sides octgon 8 sides nongon 9 sides decgon 10 sides dodecgon 12 sides REGULAR POLYGONS A regulr polygon is polygon with ll sides the sme length nd ll ngles the sme size. The polygons elow re mrked to show tht they re regulr: Equl sides re shown y smll mrkings. Equl ngles re shown using the sme symols. equilterl tringle 3 equl sides 3 equl ngles squre 4 equl sides 4 equl ngles regulr pentgon 5 equl sides 5 equl ngles regulr hexgon 6 equl sides 6 equl ngles EXERCISE 4A 1 Nme these polygons: c d e f g h

4 76 POLYGONS AND SOLIDS (Chpter 4) 2 Explin why these shpes re not polygons: c d 3 Which of the following re regulr polygons? c Angles mrked with the sme symol ² re equl in size. d e f 4 Drw the following polygons: qudrilterl with 3 equl sides n octgon with equl sides, ut with unequl ngles c hexgon with 3 right ngles. 5 Use ruler nd protrctor to determine whether the following polygons re regulr: c d

5 POLYGONS AND SOLIDS (Chpter 4) 77 B TRIANGLES A tringle is polygon with three sides. We often see tringles in structures such s uildings nd ridges ecuse they provide strength nd stility. We cn clssify tringles ccording to the numer of sides which re equl in length. A tringle is: ² sclene if the three sides ll hve different lengths ² isosceles if t lest two sides hve the sme length ² equilterl if ll three sides hve the sme length. Exmple 1 Self Tutor Clssify the following tringles: 6 cm 4 cm 7 cm 6 cm 9 cm 9 cm All three sides hve different lengths, so the tringle is sclene. Two of the sides hve the sme length, so the tringle is isosceles.

6 78 POLYGONS AND SOLIDS (Chpter 4) CONSTRUCTING A TRIANGLE We cn use compss nd ruler to construct tringle if we know the side lengths. The rdius of compss is the distnce from the shrp point to the tip of your pencil. Be creful! Your compss needle will e shrp! Compsses cn e found in the Techer Preprtion Room on top of the A3 pper wooden ox. There re two plstic oxes. Plese return the oxes fter your lesson with Grhm. Exmple 2 Self Tutor rdius Construct tringle ABC with sides 4 cm, 3 cm, nd 2 cm long. VIDEO CLIP Step 1: Drw line segment of length 4 cm. We will cll this line segment [AB], nd use it s the se of the tringle. A 4 cm B Step 2: Open your compss to rdius of 2 cm. Using this rdius, drw n rc from one end A of the se line. 2 cm A 4 cm B Step 3: Now open the compss to rdius of 3 cm. Drw n rc from B to intersect the first rc. 3 cm A 4 cm B Step 4: The point of intersection of the two rcs is the third vertex C of the tringle ABC. Drw line segments [AC] nd [BC] to complete the tringle. C 2 cm 3 cm A 4 cm B

7 POLYGONS AND SOLIDS (Chpter 4) 79 EXERCISE 4B 1 How mny tringles re in the given figures? 2 Clssify the following tringles: c 9 cm 7 cm 7 cm 6 cm 8 cm 7 cm d e f 5 cm 6 cm 4 cm 10 cm 8 cm 5 cm 6 cm 8 cm 8 cm 3 Use ruler to mesure ech side of these tringles. Hence clssify ech tringle. c d 4 Accurtely construct tringle with sides: 4 cm, 5 cm, nd 6 cm 3 cm, 6 cm, nd 7 cm. 5 Try to construct tringle with sides 3 cm, 4 cm, nd 9 cm. Is it possile to construct this tringle? Explin your nswer.

8 80 POLYGONS AND SOLIDS (Chpter 4) 6 Use protrctor nd ruler to ccurtely construct these tringles: 4 cm 55 3 cm 60 6 cm 45 7 Use compss, protrctor, nd ruler to ccurtely construct these tringles: 5 cm cm 4 cm 6 cm 8 Construct tringle ABC whose side lengths re ll 6 cm. Wht type of tringle is ABC? c Mesure the ngles of the tringle using protrctor. d Copy nd complete: All ngles of n equilterl tringle mesure... ± C QUADRILATERALS A qudrilterl is polygon with four sides. The shpes longside re ll exmples of qudrilterls. There re six specil qudrilterls: ² A prllelogrm hs oth pirs of opposite sides prllel. The opposite sides of prllelogrm re equl in length. ² A rectngle is prllelogrm with right ngled corners. The opposite sides of rectngle re equl in length. ² A rhomus is prllelogrm with ll four sides equl in length.

9 ² A squre is rectngle with ll sides equl in length. Both pirs of opposite sides of squre re prllel. POLYGONS AND SOLIDS (Chpter 4) 81 ² A trpezium hs one pir of opposite sides which re prllel. ² A kite hs two pirs of djcent sides which re equl in length. EXERCISE 4C 1 In the digrm longside, identify : squre rectngle c prllelogrm d trpezium. 2 Drw n exmple of : rhomus rectngle c trpezium d kite. 3 Nme the following qudrilterls. You my need to use ruler to mesure the sides. c d e f g h 4 Show how: two squres cn e comined to form rectngle Trpezi is the plurl of two rectngles cn e comined to form squre trpezium! c two trpezi cn e comined to form prllelogrm d two equilterl tringles cn e comined to form rhomus e two isosceles tringles cn e comined to form kite.

10 82 POLYGONS AND SOLIDS (Chpter 4) 5 True or flse? A squre is specil type of rhomus. A rectngle is specil type of squre. c A squre is specil type of prllelogrm. d A rectngle is specil type of prllelogrm. 6 Use ruler nd protrctor to drw squre with side length 6 cm. Drw the digonls of the squre. c Mesure the lengths of the digonls. Wht do you notice? 6 cm D SOLIDS Solids re ojects which occupy spce. A solid needs to e fully enclosed. However, unlike the nme suggests, it my e hollow. A piece of timer is solid, nd so is ruish in. CROSS-SECTIONS OF SOLIDS A cross-section of solid is the shpe of slice through it.

11 POLYGONS AND SOLIDS (Chpter 4) 83 If we slice this ox verticlly, the cross-section is squre. Activity 2 Cross-sections of solids Drw cross-sections of: ² lof of red ² licorice llsort ² Swiss roll ² n empty mtch ox. For some solids, the cross-section is the sme no mtter where the slice is mde. These solids re known s solids of uniform cross-section. sme different uniform cross-section not uniform cross-section PRISMS A prism is solid with uniform cross-section tht is polygon. Prisms re nmed ccording to the shpe of the cross-section. Nme Figure Cross-section Tringulr prism Rectngulr prism Hexgonl prism

12 84 POLYGONS AND SOLIDS (Chpter 4) CUBES A cue is rectngulr prism whose sides re ll the sme length. Die is the singulr of dice. A die is n exmple of cue. CYLINDERS A cylinder is solid with circulr uniform cross-section. An luminium cn is n exmple of cylinder. circle PYRAMIDS A pyrmid is solid with polygon se. It hs tringulr fces which come from its se to meet t point clled the vertex. Pyrmids re nmed ccording to the shpe of their se. vertex squre-sed pyrmid tringulr-sed pyrmid DEMO CONES A cone is solid with circulr se nd curved surfce from the se to the vertex. vertex SPHERES A sphere is ll-shped solid. cone sphere

13 POLYGONS AND SOLIDS (Chpter 4) 85 EXERCISE 4D 1 Nme these solids: c d e f g h 2 Drw n exmple of : cue rectngulr-sed pyrmid c cone d pentgonl prism. 3 Which solid would est descrie the shpe of: refrigertor ttery c this tent? 4 Stte whether the following solids hve: A only flt surfces B only curved surfces C oth flt nd curved surfces. tringulr prism sphere c squre-sed pyrmid d cylinder 5 Nme solid which hs: only rectngulr surfces only tringulr surfces.

14 86 POLYGONS AND SOLIDS (Chpter 4) E CONSTRUCTING SOLIDS One wy to construct solid is to use net. Nets re ptterns which cn e folded long certin lines so tht we cn mke 3-dimensionl models of solids. For exmple, when this net is cut out nd folded long the dshed lines, we form cue. DEMO Activity 3 Click on the icon to otin these printle nets. Print them onto light crd, nd use them to construct : ² cue ² squre-sed pyrmid ² tringulr prism. Nets PRINTABLE NETS EXERCISE 4E 1 Drw nd nme the solids which would e formed from these nets. PRINTABLE NETS 2 Drw net for ech of the following solids: tringulr-sed pyrmid hexgonl prism. 3 Drw the net which could e used to construct ox like this one: How would you chnge this net so tht the ox is open t the top?

15 POLYGONS AND SOLIDS (Chpter 4) 87 4 Mtch the net given in the first column with the correct solid nd the correct nme. Net Solid Nme A 1 pentgonl-sed pyrmid B 2 cylinder c C 3 tringulr prism d D 4 squre-sed pyrmid 5 Three students were sked to drw net for squre-sed pyrmid. The nets tht were drwn re shown elow: Clire Derek Eric Explin why it is not possile to construct pyrmid from Clire s net. Which of the remining nets will produce higher pyrmid? Explin your nswer.

16 88 POLYGONS AND SOLIDS (Chpter 4) 6 Which of these nets cn e used to mke cue? c DEMO d e f 7 Drw n exct net which could e used to construct: 1 cm 3 cm 2 cm 3 cm 1 cm Activity 4 You will need: plstic strws, plsticine Wht to do: Using strws s the edges, nd plsticine to hold the edges together, crete the following solids: ² cue ² rectngulr prism ² tringulr prism ² squre-sed pyrmid ² tringulr-sed pyrmid Models of solids We cll this wirefrme model ecuse it only includes the edges. Experiment using strws of different lengths. For ech solid, determine which edges must e the sme length, nd which edges cn e different lengths. KEY WORDS USED IN THIS CHAPTER ² cone ² cross-section ² cue ² cylinder ² equilterl ² isosceles ² kite ² net ² prllelogrm ² polygon ² prism ² pyrmid ² qudrilterl ² rectngle ² regulr polygon ² rhomus ² sclene ² solid ² sphere ² squre ² trpezium

17 POLYGONS AND SOLIDS (Chpter 4) 89 Review set 4 1 Nme the following polygons: c 2 Drw the following polygons: isosceles tringle regulr hexgon c rhomus 3 Clssify the following tringles: 6 cm c 6 cm 7 cm 10 cm 10 cm 8 cm 8 cm 8 cm 8 cm 4 Using compss nd ruler only, construct n isosceles tringle with se length 5 cm nd equl sides of length 4 cm. 5 Nme these qudrilterls. You my need to use ruler to mesure the sides. c d 6 Drw: tringulr prism cylinder c cue. 7 Nme these solids: c

18 90 POLYGONS AND SOLIDS (Chpter 4) 8 Drw nd nme the solids which would e formed from the following nets: c d Prctice test 4A Click on the icon to otin this printle test. Multiple Choice PRINTABLE TEST Prctice test 4B Short response 1 Using ruler nd protrctor, determine whether these polygons re regulr: 2 Using compss nd ruler only, construct tringle with sides of length 3 cm, 4 cm, nd 6 cm. 3 Using protrctor nd ruler, ccurtely construct tringle with the mesurements shown. 3 cm Mesure the length of the third side in mm cm

19 POLYGONS AND SOLIDS (Chpter 4) 91 4 Clssify this tringle y mesuring its sides: 5 Drw qudrilterl which hs 3 otuse ngles. 6 Explin whether: n equilterl tringle is lso isosceles squre is specil type of kite. 7 Wht solid would est descrie the shpe of: mrle filing cinet? 8 Drw the net for hexgonl-sed pyrmid. Prctice test 4C Extended response 1 Wht type of qudrilterls re those shown elow? S T X A R U W B Y Z Use your protrctor to mesure the ngles in ech shpe. c Copy nd complete: The opposite ngles of prllelogrm re...

20 92 POLYGONS AND SOLIDS (Chpter 4) 2 Answer the Opening Prolem on pge Construct tringle with side lengths: i 3 cm, 6 cm, 7 cm ii 4 cm, 5 cm, 7 cm Use your protrctor to mesure the ngles of ech tringle in. Give your nswer to the nerest degree. c Find the sum of the ngles in ech tringle. d Copy nd complete: The sum of the ngles in tringle is... 4 Four students were sked to drw net for pentgonl prism. Their responses re shown elow: Judith Nicole Chrlie Hrvey c Explin why it is not possile to construct pentgonl prism from Nicole s net. There is nother net which cnnot e used to construct pentgonl prism. Which one is it? Which of the two remining nets will produce tller prism when plced on its pentgonl end?