Geometry: similarity and mensuration

Transcription

1 Geometry: similrity nd mensurtion 8 VCEcoverge Are of study Units & Geometry nd trigonometry In this ch chpter 8A Properties of ngles, tringles nd polygons 8B Are nd perimeter 8C Totl surfce re 8D Volume of prisms, pyrmids nd spheres 8E Mps nd similr figures 8F Similr tringles 8G Are nd volume scle fctors

2 8 Further Mthemtics Geometry Geometry is n importnt re of study. Mny professions nd tsks require nd use geometricl concepts nd its ssocited techniques. Besides rchitects, surveyors nd nvigtors, ll of us use it in our dily lives for exmple, to descrie shpes of ojects, directions on cr trip nd spce or position of house. Much of this re of study is ssumed knowledge gined from previous yers of study. Properties of ngles, tringles nd polygons In this module, we will often encounter prolems where some of the informtion we need is not clerly given. To solve the prolems, some missing informtion will need to e deduced using the mny common rules, definitions nd lws of geometry. Some of the more importnt rules re presented in this chpter. Interior ngles of polygons For regulr polygon (ll sides nd ngles re equl) of 60 n sides, the interior ngle is given y n For exmple, for squre the interior ngle is: = = The exterior ngle is given y n WORKED Exmple Find the interior nd exterior ngle of the regulr polygon shown. Bed Bed Bed Stirwys Interior ngle UPPER LEVEL Bed Exterior ngle This shpe is regulr pentgon, 5-sided figure. Sustitute n = 5 into the interior ngle formul. Sustitute n = 5 into the exterior ngle formul. Write your nswer. 60 Interior ngle = = 80 7 = Exterior ngle = = 7 A regulr pentgon hs n interior ngle of 08 nd n exterior ngle of 7.

3 Chpter 8 Geometry: similrity nd mensurtion 9 Geometry rules, definitions nd nottion rules The following geometry rules nd nottion will e most vlule in estlishing unknown vlues in the topics covered nd revised in this module. Definitions of common terms B A ABC Less thn 90 Acute ngle Between 80 nd 60 Reflex ngle C MQ F 90 Right ngle M t fi 05() 80 Stright ngle MQ FurMt fig.05(c) Between 90 nd 80 Otuse ngle MQ F rmt fig 05(d) A B AB Line A B AB Ry A B AB Line segment Prllel lines Perpendiculr lines MQ FurMt fig.05(h) Some common nottions nd rules + + c = 80 No equl sides Two equl sides nd ngles All equl sides nd 60 ngles 5 c Sclene tringle MQ FurMt fig.06() Isosceles tringle MQ FurMt fig.06() Equilterl tringle 5 Right-ngled isoceles tringle + = 90 Complementry ngles MQ FurMt fig.06(e) + = 80 Supplementry ngles MQ FurMt fig.06(f) = Verticlly opposite ngles MQ FurMt fig.06(g) A C B D CD is perpendiculr isector of AB = c = d c d Alternte ngles c d + + c + d = 60 MQ FurMt fig.07(d) = c = d c d Corresponding ngles B + = d c d A C D BCD is n exterior ngle MQ FurMt fig 07(e) + d = 80 + c = 80 c d Co-interior ngles Right ngle t the circumference in semicircle

4 50 Further Mthemtics WORKED Exmple Find the vlues of the pronumerls in the polygon t right. d cm c This shpe is regulr hexgon. The ngles t the centre re ll equl. 60 = = 60 The other two ngles in the tringle re equl. + + c = 80 = c So: = 80 = 60 c = 60 The 6 tringles re equilterl tringles, therefore ll sides re equl. d cm = 6 cm 6 cm WORKED Exmple Find the missing pronumerls in the digrm of rilings for set of stirs shown t right. c Recognise tht the top nd ottom of the stir rils re prllel lines. c To find the unknown ngle, use the lternte ngle lw nd the given ngle. The unknown ngle c is right ngle, using the given right ngle nd corresponding ngle lw. Use the stright ngle rule to find the unknown ngle. Given ngle 5. = 5 c = 90 c + + c = = 80 = 80 5 = 55

5 rememer rememer Chpter 8 Geometry: similrity nd mensurtion 5 Properties of ngles, tringles nd polygons. Drw creful digrms.. Crefully interpret geometric nottions, for exmple from the digrm elow. Equl sides. Crefully consider geometric rules, such s isosceles tringles hve equl sides nd ngles. (Refer to the figures in the preceding section on definitions of common terms nd common nottions nd rules.) 8A Properties of ngles, tringles nd polygons WORKED Exmple Find the interior nd exterior ngles for ech of the following regulr polygons. Equilterl tringle Regulr qudrilterl c Hexgon d e Heptgon f Nongon g WORKED Exmple Find the vlue of the pronumerls in the following figures. c y c x 6 d e f c 8 cm 5 c 50 m

6 5 Further Mthemtics WORKED Exmple Find the vlue of the pronumerls in the following figures. c 0 x y 5 0 z t 6 d e f 7 m 70 c d n 8 0 Nme the regulr polygon tht hs the given ngle(s). Interior ngle of 08, exterior ngle of 7 Interior ngle of 50, exterior ngle of 0 c Interior ngle of 5, exterior ngle of 5 d Interior ngle of 0 e Exterior ngle of 0 5 Find the unknown pronumerls. r c 8 cm 0 d 9 x y z 5 h.6 cm d e cm c d c 6 multiple choice The vlue of is closest to: A 0 B 75 C 90 D 0 50 E 50 7 multiple choice An isosceles tringle hs known ngle of 50. The lrgest possile ngle for this tringle is: A 80 B 0 C 90 D 65 E 50

7 Chpter 8 Geometry: similrity nd mensurtion 5 Are nd perimeter Much of our world is descried y re (the mount of spce enclosed y closed figure) nd perimeter (the distnce round closed figure). Some exmples re the re of house lock, the fencing of lock of lnd, the size of edroom nd the mount of pint required to cover n oject. In this section we will review the more common shpes..05 º Lot 60 65m.07 Lot m.55 Perimeter Perimeter is the distnce round closed figure. Some common rules re:. For squres, the perimter = l. For rectngles, the perimter = (l + w) Squre l 7.9 Corner lock with wide 7 m frontge $7,000 Rectngle l.8 Corner lock with expnsive.55 m frontge $5, l l w w l. Circumference (C) is the perimeter of circle. C = π rdius = πr Circumference of l cirle r WORKED Exmple Find the perimeter of the closed figure given t right (to the nerest mm). 00 mm 600 mm The shpe is composed of semicircle nd three sides of rectngle. Add together the three components of the perimeter. Write your nswer. Perimeter = circumference where -- of circumference = -- πr = π 50 = 7. Perimeter = = 97. Perimeter of the closed figure is 97 mm. --

8 5 Further Mthemtics Are of common shpes The res of shpes commonly encountered re:. Are of squre: A = length = l Squre l. Are of rectngle: A = length width = l w. Are of prllelogrm: A = se height = h l Rectngle w l Prllelogrm h. Are of trpezium: A = -- ( + ) h 5. Are of circle: A = π rdius = π r Trpezium h Circle r O 6. Are of tringle: A = -- h (see the next chpter) Tringle Are is mesured in mm, cm, m, km nd hectres. hectre = 00 m 00 m = m 5.7 m Find the re of the grden ed given in the digrm (to the nerest squre metre). 7.5 m The shpe of grden is trpezium. = 7.5 = 5.7 h =. Use the formul for re of trpezium. Are of grden = Are of trpezium Rememer tht the lengths of the two prllel = -- ( + ) h sides re nd nd h is the perpendiculr distnce etween the two prllel sides. Sustitute nd evlute. = -- ( ). WORKED Exmple Write your nswer. 5 = --.. = 5.8 m Are of the grden ed is pproximtely 6 squre metres. h. m

9 Chpter 8 Geometry: similrity nd mensurtion 55 Composite res Often closed figure cn e identified s comprising two or more different common figures. Such figures re clled composite figures. The re of composite figure is the sum of the res of the individul common figures. WORKED Exmple Are of composite figure = sum of the individul common figures A composite = A + A + A + A Find the re of the hotel foyer from the plns given elow (to the nerest squre metre). 5 m 0 m 8 m The shpe is composite nd needs to e seprted into two or more common 5 m shpes: in this cse, rectngle, A A A tringle nd hlf of circle. 0 m 6 m 6 m 8 m Are of foyer = A + A + A Sustitute nd evlute ech of the shpes. The width of the rectngle nd the se of the tringle is twice the rdius of the circle; tht is, 6 metres. Add together ll three res for the composite shpe. A = re of tringle = -- h = = 60 m A = Are of rectngle = l w = 5 6 = 00 m A = Are of hlf of circle = -- π r = -- π 8 = 00.5 m Are of foyer = A + A + A = = m Write your nswer. Are of the hotel foyer is 66 m.

10 56 Further Mthemtics Conversion of units of re Often the units of re need to e converted, for exmple from cm to m nd vice vers.. To convert to smller units, for exmple m to cm, multiply ( ).. To convert to lrger units, for exmple, mm to cm, divide ( ). Some exmples re: () cm = 0 mm 0 mm = 00 mm () m = 00 cm 00 cm = cm (c) km = 000 m 000 m = m (d) hectre = m WORKED Exmple 7 Convert. m to squre centimetres (cm ). Are mm cm m km Conversion fctor for metres to centimetres is multiply y 00. Tht is, metre = 00 centimetres. Conversion fctor for metres to centimetres is multiply y 00 or Write your nswer.. m =. metre metre =. 00 cm 00 cm = 00 cm. m is equl to 00 squre centimetres (cm ). WORKED Exmple Convert metres to kilometres hectres. 8 Conversion fctor for metres to kilometres is divide y 000; tht is, metre = kilometre. Conversion fctor for metres to kilometres is divide y 000 or m = km km = = 0.56 km Write the nswer in correct units m = 0.56 squre kilometres (km ) Conversion fctor is m = hectre; tht is, m = hectre Write the nswer m = = hectres = 5.6 hectres m = 5.6 hectres hectres

11 Chpter 8 Geometry: similrity nd mensurtion 57 rememer rememer Are nd perimeter. Perimeter is the distnce round closed figure. () For squres, the perimeter = l Squre l l l () For rectngles, the perimeter = (l + w) (c) Circumference (C) is the perimeter of circle. C = π rdius. Are is mesured in mm, cm, m, km nd hectres.. () cm = 0 mm 0 mm = 00 mm () m = 00 cm 00 cm = cm (c) km = 000 m 000 m = m (d) hectre = m. Are of shpes commonly encountered re: () re of squre: A = l () Are of rectngle: A = l w (c) Are of prllelogrm: A = h -- l Rectngle l (d) Are of trpezium: A = ( + ) h (e) Are of circle: A = πr (f) Are of tringle: A = -- h 5. Are of composite figure = sum of the individul common figures A composite = A + A + A + A +... w Circumference l of r circle w 8B Are nd perimeter WORKED Exmple 5 Find the res of the following figures (to the nerest whole units). m c 5 m 7 m m 7.8 cm.7 cm 7.5 cm 5. cm Are nd perimeter Mthcd d e f 5 m 7.5 m.5 mm m.5 m 0 m 70 m 0 m SkillSHEET m

12 58 Further Mthemtics WORKED Exmple WORKED Exmple 6 Find the perimeters of the closed figures in question. Find the res of the following figures (to deciml plce). c 0 m m m 5 m.5 m 7 m m 0 m d e f 8 mm 90 mm mm 5 mm cm 0 cm 6 cm 8 cm 0 cm m m m m m m cm 7 m 0 m SkillSHEET 8. WORKED Exmple 7, 8 Find the perimeters of the closed figures in question. 5 Convert the following res to the units given in rckets mm (cm ) cm (m ) c 0.05 m (cm ) d 0.05 m (mm ) e m (km ) f m (hectres) g mm (m ) h km (m ) 6 A kite hs the dimensions in the figure t right. Find the re of the kite (to the nerest cm ). 70 cm 80 cm.0 m 7 Find the re of the regulr hexgon s shown in the digrm t left (to deciml plces, in m )..08 m 8 A cutting lde for crft knife hs the dimensions shown in the digrm. Wht is the re of steel in the lde? 0 mm 0 mm 5 mm 0 mm

13 Chpter 8 Geometry: similrity nd mensurtion 59 9 multiple choice The re in m of the stcked ojects shown t right is closest to: A. B.68 C.9 D.8 E m 0.8 m.0 m. m 0.8 m 0.8 m 0.8 m 0.8 m 0 multiple choice The perimeter of the figure shown in centimetres is: A B + 5π C +.5π D 9 + 5π E 9 +.5π 7 cm cm cm cm multiple choice The perimeter of the enclosed figure shown is 56.6 metres. The unknown length, x, is closest to: A 0.5 m B 5. m C 0. m D 80. m E Cnnot e determined A -ring drtord hs dimensions s shown elow left. (Give ll nswers to deciml plce.) 5. m 0.5 m x MQ FurMt fig.59 c d 0 cm 0 cm 6 cm Wht is the totl re of the drtord? Wht is the re of the ullseye (inner circle)? Wht is the re of the -point middle ring? Express ech re of the three rings s percentge of the totl re (to deciml plces). SkillSHEET 8.

14 60 Further Mthemtics On western movie set, horse is tied to riling outside sloon r. The riling is metres long; the led on the horse is lso metres long nd tied t one of the ends of the riling. Drw digrm of this sitution. To how much re does the horse now hve ccess (to deciml plce)? The led is now tied to the centre of the riling. c Drw digrm of this sitution. d To how much re does the horse hve ccess (to deciml plce)? The rectngulr rer window of cr hs n re of.8 m. Find the height of the rer window if its length is 60 centimetres (to the nerest centimetre). A wiper lde is 50 cm long nd the end just reches the top of the window s it mkes semicirculr sweep. The se of the wiper is situted t the ottom centre of the rer window. Drw digrm of the sitution. c Find the re of the window tht is swept y the wiper (to the nerest cm ). d Find the percentge of the window s re tht is not swept y the wiper. The mnufcturer decides to increse the wiper length y 0 cm. e Find the new re of the rer window tht is swept. f Find the percentge of the window s re tht is not swept y the wiper. 5 A signwriter chrges his clients y the width nd height of the sign to e pinted. A client dvises the signwriter to pint words with 0 cm high chrcters nd 0 cm length for ech word. Wht is the re of ech word? Wht re ll the different wys of rrnging the words in rectngulr pttern? c If the chrge is $ per 0 cm in height nd $.50 per 0 cm in length, find the minimum cost for the sign nd its dimensions.

15 Chpter 8 Geometry: similrity nd mensurtion 6 Totl surfce re The totl surfce re (TSA) of solid oject is the sum of the res of the surfces. In some cses, we cn use estlished formuls of very common everydy ojects. In other situtions we will need to derive formul y using the net of n oject. Totl surfce re formuls of common ojects Cue l Cuoid h l Cylinder r w h Cues: TSA = 6l Cuoids: TSA = (lw + lh + wh) r Cone h s Slnt height Sphere Cylinders: TSA = πr(r + h) r r Cones: Spheres: TSA = πr(r + s) where TSA = πr s is the slnt height WORKED Exmple 9 Find the totl surfce re of poster tue with length of. metres nd rdius of 5 cm. Give your nswer to the nerest 00 cm.. m A poster tue is cylinder. Express ll dimensions in centimetres. Rememer metre = 00 centimetres. Sustitute nd evlute. Rememer BODMAS. Write your nswer. 5 cm Rdius, r = 5 cm Height, h =. m = cm TSA of tue = πr(r + h) = π 5(5 + ) = π 5 8 = The totl surfce re of poster tue is pproximtely 700 cm.

16 6 Further Mthemtics Find the totl surfce re of size 7 sketll with dimeter of 5 cm. Give your nswer to the nerest 0 cm. WORKED Exmple Use the formul for the totl surfce re of sphere. Use the dimeter to find the rdius of the sketll nd sustitute into the formul. 0 Dimeter = 5 cm Rdius =.5 cm TSA of sphere = πr = π.5 = Write your nswer. Totl surfce re of the ll is pproximtely 960 cm. A die used in ord gme hs totl surfce re of 50 mm. Find the liner dimensions of the die (to the nerest millimetre). A die is cue. We cn sustitute into the totl surfce re of cue to determine the dimension of the cue. Divide oth sides y 6. Tke the squre root of oth sides to find l. Write your nswer. TSA = 6 l TSA = 50 mm 50 = 6 l l = = 5 l = 5 = 5 mm WORKED Exmple The dimensions of the die re: 5 mm 5 mm 5 mm Totl surfce re using net If the oject is not common oject or vrition of one, such s n open cylinder, then it is esier to generte the formul from first principles y constructing net of the oject. A net of n oject is plne figure tht represents the surfce of -dimensionl oject. Squre pyrmid Slnt height Net MQ FurMt fig.68 Net MQ FurMt fig.69 Net

17 0 cm 0 cm 0 cm Chpter 8 Geometry: similrity nd mensurtion 6 WORKED Exmple Find the totl surfce re of the tringulr prism shown in the digrm. 6 cm 0 cm 0 cm 8 cm Form net of the tringulr prism, trnsferring ll the dimensions to ech of the sides of the surfces. 0 cm 0 cm 8 cm A 6 cm A A A 0 cm 8 cm 0 cm A 6 cm 6 cm Identify the different-sized common figures nd set up sum of the surfce res. The two tringles re the sme. TSA = A + A + A + A A = l w = 0 0 = 00 cm A = l w = 0 8 = 60 cm A = l w = 0 6 = 0 cm A = -- h = = cm Sum the res. TSA = A + A + A + A = = 58 cm Write your nswer. The totl surfce re of the tringulr prism is 58 cm.

18 6 Further Mthemtics WORKED Exmple Find the surfce re of n open cylindricl cn tht is cm high nd 8 cm in dimeter (to deciml plce). cm 8 cm Form net of the open cylinder, πr trnsferring ll the dimensions to ech of the surfces. A cm A cm Identify the different-sized common figures nd set up sum of the surfce res. The length of the rectngle is the circumference of the circle. TSA = A + A A = πr w = π = 0.59 cm A = π r = π = 50.7 cm Sum the res. TSA = A + A = = 5.86 cm Write your nswer. The totl surfce re of the open cylindricl cn is 5.9 cm. rememer rememer Totl surfce re. Totl surfce re (TSA) is mesured in mm, cm, m nd km.. The TSAs of some common ojects re s follows: () Cues: TSA = 6l () Cuoids: TSA = (lw + lh + wh) (c) Cylinders: TSA = πr(r + h) (d) Cones: TSA = πr(r + s) where s is the slnt height (e) Spheres: TSA = πr. For ll other shpes, form their nets nd estlish the totl surfce re formuls.

19 Chpter 8 Geometry: similrity nd mensurtion 65 8C Totl surfce re WORKED Exmple 9 WORKED Exmple 0 Find the totl surfce re for ech of the solids to f from the following informtion. Give nswers to deciml plce. A cue with side lengths of 0 cm A cuoid with dimensions of m 5 m 8 m (l w h) c A sphere with rdius of 0.8 metres d A closed cylinder with rdius of. cm nd height of 6 cm e A closed cone with rdius of 7 cm nd slnt height of cm f An opened cylinder with dimeter of 00 mm nd height of 0 mm Find the totl surfce re of the ojects given in the digrms. Give nswers to deciml plce. Length =.5 m c Totl surfce re Mthcd 0 mm cm Dimeter = cm 7 cm cm d e f 6 cm 90 cm cm 8 cm cm 8 cm WORKED Exmple WORKED Exmple Find the unknown dimensions, given the totl surfce re of the ojects. Give nswers to deciml plce. Length of cue with totl surfce re of m The rdius of sphere with totl surfce re of 6.5 cm c Length of cuoid with width of mm, height of 6 cm nd totl surfce re of 68 cm d Dimeter of plying ll with totl surfce re of cm Find the totl surfce res for the ojects given in the digrms. Give nswers to deciml plce. cm 5 cm 0 cm 5 cm 6.06 cm 7 cm 0 cm

20 cm 66 Further Mthemtics c Are = cm d 8 cm mm 0 mm 80 mm 0 mm 05 mm cm e f 5 m m WORKED Exmple mm 5 mm 9 mm 5 mm 5 Find the totl surfce re of ech of the ojects in the digrms elow. Give nswers to deciml plce. Ruish in m 50 mm 0.5 cm 6 m 0 cm 7 m 0 cm.5 cm 5 cm 0 m c 50 mm. m.5 m.5 m 0.9 m 6 A concrete swimming pool is cuoid with the following dimensions: length of 6 metres, width of metres nd depth of. metres. Wht surfce re of tiles is needed to line the inside of the pool? (Give nswer in m nd cm to deciml plce.) d.5 cm 7 cm cm

21 Chpter 8 Geometry: similrity nd mensurtion 67 7 Wht is the totl re of cnvs needed for the tent (including the se) shown in the digrm t right? Give the nswer to deciml plces..0 m.5 m.5 m.5 m 6.5 m 8 multiple choice The totl surfce re of 8 mm-dimeter ll used in gme of pool is closest to: A 80 mm B 00 mm C 70 mm D mm E mm 9 multiple choice The totl surfce of golf ll of rdius mm is closest to: A 550 mm B 55 cm C mm D m E 5.5 cm 0 multiple choice The formul for the totl surfce re for the oject shown is: A -- h B -- h + + h C ( -- h + ) h D -- h + E h + multiple choice The totl surfce re of poster tue tht is 5 cm long nd 8 cm in dimeter is closest to: A 000 cm B 900 cm C 500 mm D 600 m E 000 cm A ker is investigting the est shpe for lof of red. The shpe with the smllest surfce re stys freshest. The ker hs come up with two shpes: rectngulr prism with cm-squre se nd cylinder with round end tht hs cm dimeter. Which shpe stys fresher if they hve the sme overll length of cm? Wht is the difference etween the totl surfce res of the two loves of red? WorkSHEET 8.

22 68 Further Mthemtics Volume of prisms, pyrmids nd spheres The most common volumes considered in the rel world re the volumes of prisms, pyrmids, spheres nd ojects which re comintion of these. For exmple, country people who rely on tnk wter need to know the cpcity (volume) of wter tht the tnk is holding. Volume is the mount of spce occupied y -dimensionl oject. The units of volume re mm (cuic millimetres), cm (cuic centimetres or cc), nd m (cuic metres). 000 mm = cm cm = m Another mesure of volume is the litre which is used primrily for quntities of liquids ut lso for cpcity, like the cpcity of refrigertor, or the size of motor cr engines. litre = 000 cm 000 litres = m Conversion of units of volume Often the units of volume need to e converted, for exmple from cm to m nd vice vers. Volume 0 00 mm cm m WORKED Exmple Convert. cm to mm. The conversion from centimetres to millimetres is cm = 0 mm cm =. cm cm cm =. 0 mm 0 mm 0 mm The conversion fctor for cm to mm is to multiply y 0 or 000; tht is, cm = 000 mm. =. 000 mm = 0 mm Write the nswer in correct units.. cm is equl to 0 mm.

23 Chpter 8 Geometry: similrity nd mensurtion 69 WORKED Exmple 5 Convert cm to: m litres. The conversion fctor for centimetres to metres is divide y 00; tht is, cm = m. 00 The conversion fctor for cm to m is divide y 00 or ; tht is, cm = m cm = cm cm cm = m m m = m = 0.56 m 00 Write the nswer in correct units cm = 0.56 cuic metres (m ) Conversion fctor is 000 cm = litre; tht is, cm = litre cm = litres = litres = 56 litres 000 Write the nswer cm = 56 litres Volume of prisms A prism is -dimensionl oject tht hs uniform cross-section. Tringulr prism Cylinder Squre prism A prism is nmed in ccordnce with its uniform crosssectionl re. Note: Circulr prisms re clled cylinders. To find the volume of prism we need to determine the re of the uniform cross-section (or se) nd multiply y the height. This is the sme for ll prisms. Uniform cross-section Height Volume of prism, V prism, cn e generlised y the formul: V prism = re of uniform cross-section height V = A H

24 70 Further Mthemtics Find the volume of the oject (to the nerest cm ). The oject hs circle s uniform cross-section. It is cylinder. The re of the se is: re of circle = πr. Volume is cross-sectionl re times height. WORKED Exmple 6 V cylinder = A H where A circle = πr 5 cm V cylinder = π r H = π 5 0 = 500 π = cm Write your nswer. The volume of the cylinder is 7 cm. 0 cm WORKED Exmple 7 Find (to the nerest mm ) the volume of the slice of red with uniform cross-sectionl re of 50 mm nd thickness of 7 mm. Are 50 mm 7 mm The slice of red hs uniform crosssection. The re of the cross-section is not common figure ut its re hs een given. V = A H where A = 50 mm V = 50 mm 7 mm = 50 mm Write your nswer. The volume of the slice of red is 50 mm. Given the volume of n oject, we cn use the volume formul to find n unknown dimension of the oject y trnsposing the formul.

25 Chpter 8 Geometry: similrity nd mensurtion 7 WORKED Exmple 8 Find the height of the tringulr prism from the informtion provided in the digrm t right (to deciml plce). Volume of prism = 6.6 m h m. m The volume of the oject is given, long with the width of the tringulr cross-section nd the height of the prism. Sustitute the vlues, trnspose nd evlute. Write your nswer. V = 6.6 m, H =. m, = m V = A H where A = -- h V = h H = -- h. =. h 6.6 h = The height of the tringle in the given prism is 6.0 metres. Volume of pyrmids A pyrmid is -dimensionl oject tht hs similr cross-section ut the size reduces s it pproches the vertex. Vertex Tringulr pyrmid The nme of the pyrmid is relted to its similr cross-sectionl re (or se). Note: Circulr pyrmids re commonly clled cones. To find the volume of the pyrmids ove, we tke similr pproch to prisms ut the volume of pyrmid is lwys one-third of prism with the sme initil se nd sme height, H. This is the sme for ll pyrmids. Cone Volume of pyrmid, V pyrmids, cn e generlised y the formul: V pyrmids = re of cross-section t the se height V = A H The height of pyrmid, H, is sometimes clled the ltitude. H A

26 7 Further Mthemtics WORKED Exmple 9 Find the volume of the pyrmid t right (to the nerest m ). Height of pyrmid = 0 m The pyrmid hs squre se. It is squre pyrmid. The re of the se is: Are of squre = l. V pyrmid = -- A H where A squre = l V pyrmid = -- l H = m 0 m = 000 m Write your nswer. The volume of the squre pyrmid is 000 m. Volume of spheres nd composite ojects Volume of sphere Spheres re unique ut common ojects tht deserve specil ttention. The formul for the volume of spheres is: V sphere = -- πr where r is the rdius of the sphere. r

27 Chpter 8 Geometry: similrity nd mensurtion 7 Volume of composite ojects Often the oject cn e identified s comprising two or more different common prisms, pyrmids or spheres. Such figures re clled composite ojects. The volume of composite oject is found y dding the volumes of the individul common figures or deducting volumes. The grin silo cn e modelled s the sum of cylinder nd lrge cone, less the tip of the lrge cone. Volume of composite oject = sum of the individul common prisms, pyrmids or spheres. V composite = V + V + V +... (or V composite = V V ) WORKED Exmple 0 Find the volume of the oject shown t right (to the nerest litre). cm 0 cm The oject is composite of cylinder nd squre prism. r = 6 cm The volume of the composite oject is the sum of volumes of the cylinder plus the prism. Convert to litres using the conversion of 000 cm = litre. Write your nswer. H = 0 cm 8 cm 5 cm 8 cm 8 cm V composite = volume of cylinder + volume of squre prism = A circle H circle + A squre H squre = (πr H c ) + (l H s ) = (π 6 0) + (8 5) = = cm 0 6 cm = 0.6 litres The volume of the oject is 0 litres. 5 cm

28 7 Further Mthemtics rememer rememer Volume of prisms, pyrmids nd spheres. Volume is the mount of spce occupied y -dimensionl oject.. () The units of volume re mm, cm (or cc), m. () 000 mm = cm (c) cm = m (d) litre = 000 cm (e) 000 litres = m. The volume of prism is V prism = re of uniform cross-section height V = A H. () The volume of pyrmid is V pyrmid = -- re of cross-section t the se height V = A H () The height of pyrmid, H, is sometimes clled the ltitude. 5. The volume of sphere is V sphere = -- πr. 6. The volume of composite oject = sum of the individul common prisms, pyrmids or spheres. V composite = V + V + V +... (or V composite = V V... ) -- 8D Volume of prisms, pyrmids nd spheres Mthcd SkillSHEET Volume formuls 8. WORKED Exmple, 5 WORKED Exmple 6 Convert the volumes to the units specified. 0.5 cm to mm 800 cm to m c cm to litres d 5 litres to cm e.6 m to litres f 0.00 cm to mm g m to cm h mm to litres i mm to cm Find the volume of the following prisms to the nerest whole unit. c 7 cm 75 mm 5. cm 000 mm 0.8 cm cm cm d e f. m 6. m.8 m 0 mm 0 mm mm mm 5 cm mm 57 mm

29 Chpter 8 Geometry: similrity nd mensurtion 75 WORKED Exmple 7 Find the volume of the following prisms (to deciml plces). Are =. m 0.5 m c Are = 0 mm.9 m d Are = 000 cm Are = 5 cm.5 mm WORKED Exmple 8 Are = cm 8.5 cm Find the mesurement of the unknown dimension (to deciml plce). Volume of cue Volume of tringulr prism = c x d =.78 m 6. cm x x 5.0 cm 8 Volume of prism = 0 litres. cm 0 mm Volume of cylinder = mm x x WORKED Exmple 9 5 Find the volume of these pyrmids (to the nerest whole unit). V c VO = 7m VO = 0 cm 5 cm V cm 8 m O m O cm cm d e mm f cm O V V cm VO = 8 cm Altitude of squre pyrmid = 8 mm O Bse of pyrmid VO = 5 cm 6 cm 6 cm 0 cm

30 76 Further Mthemtics WORKED Exmple 0 6 Find the volume of these ojects (to the nerest whole unit). cm c r = 8 cm 7 cm 8 cm 0 cm m 5 m 0 cm m d e f m 6 m m m m 0 cm 0 cm 5 cm. m.5 m 6 m g h 9 m m m 60 m 00 mm 5 mm 7 Find the volume of cue with sides.5 cm long. Find the volume of room,.5 m y m y. m high. c Find the rdius of sell tht hs volume of 5 cm. d e f g h Find the volume of squre pyrmid, cm squre nd 0 cm high. Find the height of cylinder tht is 0 cm in dimeter with volume of.5 litres (to the nerest unit). Find the height of tringulr prism with se re of 8 mm nd volume of 0 mm. Find the depth of wter in swimming pool which hs cpcity of litres. The pool hs rectngulr dimensions of 8 metres y 5.5 metres. Find the rdius of n ice-crem cone with height of cm nd volume of 9.5 cm. 8 The medicine cup elow hs the shpe of cone with dimeter of cm nd height of 5 cm (not including the cup s se). Find the volume to the nerest millilitre, where cm = ml. cm 5 cm

31 Chpter 8 Geometry: similrity nd mensurtion 77 9 Tennis lls hve dimeter of 6.5 cm nd re pckged in cylinder tht cn hold four tennis lls. Assuming the lls just fit inside cylinder, find: the height of the cylindricl cn the volume of the cn (to deciml plce) c the volume of the four tennis lls (to deciml plce) d the volume of the cn occupied y ir e the frction of the cn s volume occupied y the lls. 0 multiple choice The volume mm is equivlent to: A litres B cm C 0 cm D 00 cm E 000 cm multiple choice The rtio of the volume of sphere to tht of cylinder of similr dimensions, s shown in the digrm, is est expressed s: A B C D E r ---- h multiple choice If the volume of the squre pyrmid shown is 6000 m, then the perimeter of the se is closest to: A 900 m V VO = 0 m B 0 m C 0 m D 80 m O E 0 m multiple choice A tin of fruit is cm high nd 0 cm in dimeter. Its volume, to deciml plce, is: A 0.0 cm B 50.5 cm C 0. cm D 00. cm E 08. cm A model eroplne is controlled y tethered string of 0 metres length. The opertor stnds in the middle of n ovl. (Give ll nswers to the nerest whole unit.) Wht is the mximum re of the ovl occupied y the plne in flight? If the plne cn e mnoeuvred in hemisphericl zone, find: i the surfce re of the irspce tht the plne cn occupy ii the volume of irspce tht is needed y the opertor for controlling the plne. c Repet prt with new control string length of 5 metres. r r

32 78 Further Mthemtics Mps nd similr figures Mps nd scles We often need to refer to mps for specifying loctions or for estlishing distnces etween two loctions. Mps re reduction of lengths in rel life; tht is, they hve the sme shpe s the originl ut re much smller in size. A mesure of the mount of reduction is the mp scle. There re two types of mp scles.. A rtio scle where, for exmple, :00 mens tht unit on the mp represents 00 units in rel life. In the mp elow one unit on the mp represents units. SCALE : METRES KILOMETRES. A simple conversion scle where, for exmple, cm = 00 m mens cm on the mp represents 00 metres in rel life. In the mp elow cm on the mp represents km. Kilometres Kilometres Converting from one type of mp scle to nother is shown in the following exmple.

33 Chpter 8 Geometry: similrity nd mensurtion 79 WORKED Exmple Convert the following mp rtio scles: : to simple conversion scle with units of centimetres :5 model scle to simple scle with units of millimetres c : to simple scle with units of centimetres. Rewrite the mp scle including the unit centimetres. : cm: cm Convert cm to more pproprite unit of length, for cm: m 00 exmple 00 cm = m. cm = 500 m Rewrite the mp scle including the unit millimetres. Divide y to reduce to unit. :5 mm = 5 mm mm =.5 mm c Rewrite the mp scle including the unit centimetres. c : cm = cm Convert cm to more pproprite unit of length. cm = m 00 Rememer 00 cm = m m = km cm = km 000 cm =.5 km To find the distnce represented on mp, use the simple conversion scle nd proportion to the desired vlue s shown in the next two exmples. Converting mp distnces to rel-life distnces WORKED Exmple Find the distnce in rel life represented y: 7 mm on mp with : scle.5 cm on mp with scle cm = 50 km. Convert mp scle rtio to conversion scle. : mm: mm mm:00 m A mp distnce of 7 mm corresponds to n ctul distnce of 7 times 00 m. 7 mm = 7 00 m 7 mm = 700 m Write your nswer. 7 mm on the mp represents 700 m in rel life. Proportion the scle y multiplying oth sides y.5. Write your nswer. cm = 50 km.5 cm =.5 50 km.5 cm = 575 km On mp with scle of cm = 50 km,.5 cm represents 575 km.

34 80 Further Mthemtics Converting rel life distnces to mp distnces On mp with mp rtio scle of :00 000, find the distnce tht would represent distnce of: 5 km 500 m. WORKED Exmple Convert rtio scle to simple conversion scle using n pproprite unit of mesure. : cm: cm cm = 000 m cm = km Multiply y.5 to go from km to cm = km 5 km nd do it with oth sides..5.5 x cm = 5 km Write your nswer. Use cm = 000 m nd divide oth sides y to go from 000 m to 500 m..5 cm = 5 km On : mp, 5 km is represented s.5 cm. cm = 000 m Convert -- or 0.5 cm to mm. Write your nswer. x cm = 500 m 0.5 cm = 500 m.5 mm = 500 m On : mp, 500 m is represented y.5 mm. Similr figures Two ojects tht hve the sme shpe ut different size re sid to e similr. For two figures to e similr, they must hve the following properties:. The rtios of the corresponding sides must e equl. A B AB B C C D A D = = = BC CD AD = common rtio. The corresponding ngles must e equl. A = A B = B C = C D = D A' B' B' 5 D' C' 6 60 C' B C A D B 5 60 C A' 85 D' A 85 D

35 Chpter 8 Geometry: similrity nd mensurtion 8 Scle fctor, k A mesure of the reltive size of the two similr figures is the scle fctor. The scle fctor is the common rtio of the corresponding sides nd quntifies the mount of enlrgement or reduction one figure undergoes to trnsform into the other figure. The strting shpe is commonly referred to s the originl nd the trnsformed shpe s the imge.. Scle fctor, k, is the mount of enlrgement or reduction nd is expressed s integers, frction or mp scle rtios. For exmple, k =, k = or : length of imge A B B C C A 9 9. Scle fctor, k = = = = length of originl AB BC CA where for enlrgements, k is greter thn nd for A C reductions, k is etween 0 nd.. For k =, the figures re exctly the sme shpe nd size nd A' C' re referred to s congruent. Enlrgements nd reductions re importnt in mny spects of photogrphy, mp mking nd modelling. Often, photogrphs re douled in size (enlrged), while house plns re n exmple of reduction to scle, for exmple :5. B B' WORKED Exmple For the similr shpes shown t right: find the scle fctor for the reduction of the shpe find the unknown length in the smll shpe. As it is reduction, the lrger shpe is the originl nd the smller shpe is the imge. 0 cm Originl 5 cm Imge 0 cm Continued over pge x

36 8 Further Mthemtics The two shpes hve een stted s eing similr, so set up the scle fctor rtio, k. Use the scle fctor to determine the unknown length s ll corresponding lengths re in the sme rtio. length of imge Scle fctor, k = length of originl A B = AB 0 cm = cm = Scle fctor, k = x = -- 5 cm x =.5 cm Write your nswers. The scle fctor of reduction is -- nd the unknown length on the smller shpe is.5 cm. WORKED Exmple 5 k = -- = length of imge length of originl x cm Prove tht the figures given elow re similr. Given tht the scle fctor is, find the lengths of the two unknown sides s nd t. s m 00m t m 0 0 m Firstly, orientte the figures to identify corresponding sides nd ngles esily. Clculte the missing ngles nd compre ech pir of corresponding ngles. s m 00m 0 0 m Imge Originl Sum of interior ngles = 60 All corresponding ngles re equl. t 50 m

37 Chpter 8 Geometry: similrity nd mensurtion 8 As the scle fctor given is for enlrgements, the originl is the smller figure. Set up the scle fctor rtio for ech of the two sides. length of imge Scle fctor, k = length of originl s For s = m s = 0 m = 60 m 70 m For t = t 70 m t = = 5 m rememer rememer Mps nd scles Mp scles cn e stted s:. A rtio scle. For exmple, :00 mens tht unit on the mp represents 00 units in rel life.. A simple conversion scle. For exmple, cm = 00 m mens cm on the mp represents 00 metres in rel life. Similr figures C' For two figures to e similr, they must hve the B' following properties:. The rtios of the corresponding sides must e equl. 6 A B B C C D A D = = = = common rtio A' AB BC CD AD D' C'. The corresponding ngles must e equl. B' 60 5 A = A B = B C = C D = D B C A D B 5 60 C A' Scle fctor, k A D'. Scle fctor, k, is the mount of enlrgement or reduction nd is expressed s integers, frctions or mp scle rtios, for exmple k =, k = or : D length of imge A B B C C A. Scle fctor, k = = = = length of originl AB BC CA where for enlrgements, k is greter thn nd for reductions, k is etween 0 nd.. For k =, the figures re exctly the sme shpe nd size nd re referred to s congruent. B' B 9 9 A C A' C'

38 8 Further Mthemtics 8E Mps nd similr figures WORKED Exmple Convert the following mp rtio scles to simple conversion scles with cm s the unit of mesure. : :000 c :5 000 d :0 000 e : f :500 Mthcd Scle fctor WORKED Exmple Stte the rel-life distnce represented on mp for ech of the following: cm on cm =.5 km mp 8.5 cm on cm = 00 m mp c 8 mm on mm = 00 m mp d cm on : mp e 7 cm on :0 000 mp f 5 mm on : mp. WORKED Exmple Stte the distnce on mp for ech of the following: km on : mp 750 m on :5 000 mp c 00 km on : mp d 5 m on :500 mp e 00 m on : mp f km on : mp. WORKED Exmple For ech of these pirs of similr shpes, find: i the scle fctor ii the vlue of x nd y. 00 cm y cm x cm metres 70 x cm m m 0 cm y cm c cm 8 cm d y mm mm 8 cm y cm x x cm cm 6 mm 7 mm mm

39 6 mm 0 mm Chpter 8 Geometry: similrity nd mensurtion 85 WORKED Exmple 5 5 Prove tht the following pirs re similr figures nd find the vlue of. 7 mm 8 mm cm cm 0 7 mm cm cm cm 5 7 cm 8 cm c d 7.5 Photo Height of person = 86 cm 0 6 A photo hs the dimensions 0 cm y cm. The photo is enlrged y fctor of.5. Find the new dimensions of the photo. 7 Most scle model crs re in the rtio :. Find: the length of rel cr if the model is 0 cm long (in metres to deciml plce) the height of rel cr if the model is cm high (to the nerest centimetre) c the length of model if the rel cr is metres long. 8 The dimensions of student s room re 00 mm y 560 mm. An pproprite scle to drw scle digrm the scle drwing of the room nd stte whether on n A sheet is :0. N Thoms Find the dimensions of Bdger Blzing the drwing should e DAVIS LAND lndscpe or portrit on Temple the A sheet. 9 The mp t right uses Dny Prry line scle. West Clowes Convert the line scle to Se Shrpe simple conversion scle. KILOMETRES Stte the mp scle rtio c Find the stright-line distnces etween: McLeod i McLeod nd Thoms ii McLeod nd Clowes iii Shrpe nd Thoms. Gold mine Col mine Copper mine Silver mine Temple River Rome River

40 86 Further Mthemtics 0 Find the distnce etween the following pirs of loctions in the mp (to the nerest kilometre). SCALE : N 750 m Kntr Mrtin Reneton River Pxton Stuckley Se Shelly Bech Blett Goldern Se 5 m Bolivi Foster Plins BRAMBLETOWN SnkeRiver Ross Swing Jewel River Newury 0 m Mrkhm Chrleston From Brmletown to Ross in stright line From Chrleston to Mrkhm in stright line c From Shelly Bech to Blett in stright line d From Chrleston to Ross in stright line e From Chrleston to Ross vi the rods. (Hint: Use length of string to mesure the distnce.) Using the mp from question 0, stte which town(s) is/re within 5 kilometres of Brmletown. multiple choice The perimeter of the rel oject shown in the scle digrm of :5 is: A 6 cm B 5 cm C 57 cm D.8 cm E 50 cm cm multiple choice A :7 scle model of truck is mde from cly. Wht is the length of the try on the originl truck, if it is 7 cm length on the model? A cm B 00 cm C 70 cm D 50 cm E 79 cm multiple choice A scle fctor of 0. is: A reduction with scle of cm = cm B n enlrgement with scle of cm = 0. cm C n enlrgement with scle of cm = 5 cm D reduction with scle of cm = 5 cm E reduction with scle of cm = 0 cm cm

41 Chpter 8 Geometry: similrity nd mensurtion 87 Similr tringles Similr tringles cn e used to find the height of trees nd uildings or widths of rivers nd mountins. One extr rule cn e used to identify similr tringles to those mentioned for similr shpes in the previous section. Two tringles re similr if one of the following conditions is identified:. All three corresponding ngles re equl (AAA).. All three corresponding pirs of sides re in the sme rtio (liner scle fctor) (SSS). sf = -- = -- = -- = Two corresponding pirs of sides re in the sme rtio nd the included ngles re equl (SAS). 6 6 sf = -- = -- = As in the previous section, we use the known vlues of pir of corresponding sides to determine the scle fctor for the similr tringles. WORKED Exmple OA Scle fctor, k = = OA 6 For the similr tringles in the digrm, find the scle fctor the vlue of the pronumerl, x. length of side of imge length of corresponding side of originl A B C 6 B' 00 Identify tht the two tringles re similr ecuse they hve equl ngles (AAA). The third ngle is not given ut use the rule tht ll ngles in tringle sum to 80. B Originl A 6 C A' 0 x B' 6 00 Imge A' 0 50 x C' Continued over pge C'

42 88 Further Mthemtics Alwys select the tringle with the unknown length, x, s the imge. Evlute the scle fctor y selecting pir of corresponding sides from the two tringles with known lengths. Use the scle fctor to find the unknown length, x. Trnspose nd evlute. Write nswer in the correct units nd level of ccurcy. Scle fctor, length of side of imge k = length of corresponding side of originl A B = AB 6 = -- =.5 Scle fctor, k =.5 A C.5 = AC x.5 = -- 6 x =.5 6 x = 9 The scle fctor is.5 nd the unknown length, x, is 9 units. WORKED Exmple 7 For the given similr tringles, find the vlue of the pronumerl, x. D.5 B.0 E x C A 7 All mesurements in m Confirm tht the two tringles re similr ecuse they hve equl ngles (AAA). This conclusion is supported y the prllel lines shown nd using corresponding lw nd common ngle, A. A.0 m B 7 m 7.5 m Originl C D Imge E For cler nlysis seprte the two tringles. Note tht the lengths of the sides AE nd AD re the sum of the given vlues. A AD AE (7 + x) m = = 7.5 m = (7 + x) m

43 Chpter 8 Geometry: similrity nd mensurtion 89 Select s the imge the tringle with the unknown length. Evlute the scle fctor y selecting pir of corresponding sides from the two tringles with known lengths. 5 Use the scle fctor to find the unknown length. Trnspose nd evlute. Write nswer in the correct units nd level of ccurcy. Scle fctor, length of side of imge k = length of corresponding side of originl AD = AB 7.5 = k =.875 Scle fctor, length of side of imge k = length of corresponding side of originl AE.875 = AC 7 + x.875 = = 7 + x.5 = 7 + x x =.5 7 x = The vlue of x is 6 -- metres. There re mny prcticl pplictions of similr tringles in the rel world. It is prticulrly useful for determining the lengths of inccessile fetures such s the height of tll trees or the widths of rivers. This prolem is overcome y setting up tringle similr to the feture to e exmined, s shown in the next exmple. WORKED Exmple 8 Find the height of the tree shown in the digrm t right. Give the nswer to deciml plce. Shdow (0 cm) Girl (68 cm) Sun's rys Confirm tht the two tringles re similr ecuse they hve equl ngles (AAA). This conclusion is supported y prllel lines, ssuming the tree nd the girl re perpendiculr to the ground nd using corresponding lw nd common ngle, A. Originl 68 cm 0 cm m metres x m Imge Continued over pge

44 90 Further Mthemtics 5 For cler nlysis seprte the two tringles. Select the tringle with the unknown length s the imge. Evlute the scle fctor y selecting pir of corresponding sides from the two tringles with known lengths. Note: All mesurements should e in the sme units, preferly in metres. height of tree (imge) Scle fctor, k = height of girl (originl) x k = = x 0 = Trnspose nd evlute. x = 0.68 = 6.8 m Write nswer in the correct units. Height of the tree is 6.8 metres. rememer rememer Similr tringles OA. Scle fctor, k = = OA length of side of imge length of corresponding side of originl. Two tringles re similr if one of the following conditions is identified: () All three corresponding ngles re equl (AAA). () All three corresponding pirs of sides re in the sme rtio (liner scle fctor) (SSS). 6 sf = = = 6 = 0.5 (c) Two corresponding pirs of sides re in the sme rtio nd the included ngles re equl (SAS). 6 8

45 Chpter 8 Geometry: similrity nd mensurtion 9 8F Similr tringles WORKED Exmple 6 Stte the rule (SSS or AAA or SAS) tht proves the pir of tringles re similr nd determine the scle fctor (expressed s n enlrgement k > ). c mm 60 mm 0 mm 80 mm d e f WORKED Exmple 6 For the given similr tringles, find the vlue of the pronumerl,..5 mm c m m 5 cm Cri Similr tringles Geometry 5 mm mm 56 mm m. m d 7 e f x x 75 cm 5 cm x 59 cm 8 6 x WORKED Exmple 7 For the given similr tringles, find the vlue of the pronumerl,. c

46 9 Further Mthemtics d e f m 7 m 8 m 80 mm 68 m 08 mm 80 mm WORKED Exmple 8 Find the height of the flgpole shown in the digrm t right (to the nerest centimetre). Guy wire 0.9 m 5 Find the length of the ridge, AB, needed to spn the river, using similr tringles s shown (to the nerest decimetre). m 9 m B (Not to scle All mesurement re in metres).5 m A.5 m. m WorkSHEET 8. 6 The shdow of tree is metres nd t the sme time the shdow of metre stick is 5 cm. Assuming oth the tree nd stick re perpendiculr to the horizontl ground, wht is the height of the tree? 7 Find the width of the lke (to the nerest metre) using these surveyor s notes t right. 8 In the given digrm, the length of side is closest to: A B C 6 D 5 E 9.6 Questions 9 nd 0 refer to the following informtion. A young tennis plyer s serve is shown in the digrm. Assume the ll trvels in stright line. 0.9 m 9 multiple choice The height of the ll just s it is hit, x, is closest to: 5 m 0 m A.6 m B.7 m C.5 m D.8 m E.6 m 0 multiple choice 6 Not to scle 0 5 m. m multiple choice The height of the plyer, y, s shown is closest to: A 90 cm B 80 cm C 70 cm D 60 cm E 50 cm A Lke m y. m B x

47 Chpter 8 Geometry: similrity nd mensurtion 9 Are nd volume scle fctors An unknown re or volume of figure cn e found without the need to use known formuls such s in exercises 8B nd 8D. We hve seen tht two figures tht re similr hve ll corresponding lengths in the sme rtio or (liner) scle fctor, k. The sme cn e shown for the re nd volume of two similr figures. Are of similr figures If the lengths of similr figures re in the rtio : or k, then the res of the similr shpes re in the rtio : or k. Following re investigtions to support this reltionship. Different length rtios (or scle fctors) of squre length of lue squre cm = = = k length of red squre cm re of lue squre cm = = = re of red squre = k cm length of green squre length of red squre cm = = = k cm cm Are = cm cm cm Are = cm cm re of green squre re of red squre 9 cm = = 9 = = k cm cm Are = 9 cm Different length rtios (or scle fctors) of circle rdius length of lue circle rdius length of red circle re of lue circle re of red circle cm = = = k cm π cm = = = = k π cm cm cm cm Are = πr = π cm Are = πr = π cm rdius length of green circle rdius length of red circle cm = = = k cm cm Are = πr = 9π cm re of green circle re of red circle 9π cm = = 9 = = k π cm From ove, s long s two figures re similr then the re rtio or scle fctor is the squre of the liner scle fctor, k. The sme pplies for the totl surfce re. re of imge Are scle rtio or fctor (sf) = re of originl = squre of liner scle fctor (lsf) = (lsf) = k

48 9 Further Mthemtics The steps required to solve for length, re or volume (investigted lter) using similrity re:. Clerly identify the known corresponding mesurements (length, re or volume) of the similr shpe.. Estlish scle fctor (liner, re or volume) using known mesurements.. Convert to n pproprite scle fctor to determine the unknown mesurement.. Use the scle fctor nd rtio to evlute the unknown. For the similr tringles shown, find the re, x cm, of the smll tringle. 5 WORKED Exmple length of smll tringle (imge) Determine scle fctor, in this Liner scle fctor = length of lrge tringle (originl) instnce the liner scle fctor, from the two corresponding lengths. cm k = given. It is preferred tht the.8 cm unknown tringle is the imge. = -- Determine the re scle fctor. Are scle fctor = k Use the re scle fctor to find the unknown re. 9 = = Are scle fctor = Are = x. cm -- x cm -- = cm Trnspose the eqution to get x = unknown y itself. x = 5 Write your nswer. The re of the smll tringle is 5 cm. For the two similr shpes shown, find the unknown length, x cm. WORKED Exmple Determine scle fctor, in this instnce the re scle fctor, s oth res re known. It is preferred tht the tringle with the unknown is stted s the imge Are = 00 cm.8 cm re of smll tringle (imge) re of lrge tringle (originl) cm 0 cm 50 cm re of imge (lrge trpezium) Are scle fctor = re of originl (smll trpezium) k 50 cm = cm = 5 x

49 Chpter 8 Geometry: similrity nd mensurtion 95 Determine the liner scle fctor. k 5 Use the liner scle fctor to find the unknown length. Trnspose the eqution to get unknown y itself. Write your nswer. Liner scle fctor = k = 5 k = 5 length of imge (lrge trpezium) Liner scle fctor = length of originl (smll trpezium) x cm 5 = cm x = 5 x = 0 The length, x, is 0 cm. Volume of similr figures If the lengths of similr figures re in the rtio : or k, then the volume of the similr shpes re in the rtio : or k. The following is n investigtion of two different ojects, cues nd rectngulr prisms. A cue length of lrge (lue) cue length of smll (red) cue cm = = = k cm Volume = = cm cm cm cm volume of lrge cue volume of smll cue 8 cm = = 8 = = k cm Volume = = 8 cm A rectngulr prism length of smll prism cm = = -- = k length of lrge prism 6 cm volume of smll prism cm = = -- = -- volume of lrge prism cm 8 = k From ove, s long s two figures re similr then the volume rtio or scle fctor is the cue of the liner scle fctor, k. volume of imge Volume scle fctor (vsf) = volume of originl = cue of liner scle fctor (lsf) = (lsf) = k cm Volume = = cm cm cm Volume = 6 = cm cm cm cm cm 6 cm cm

50 96 Further Mthemtics WORKED Exmple For the two similr figures shown, find the volume of the smller cone. Volume of lrge cone = 50 cm 6 cm 9 cm Seprte the two figures to clrify the detils of the similr figures. Volume = 50 cm 6 cm 9 cm Volume = x cm Determine scle fctor, in this instnce the liner scle fctor, from the two corresponding lengths given. It is preferred tht the unknown tringle is the imge. Determine the volume scle fctor. Use the volume scle fctor to find the unknown length. length of smll tringle (imge) Liner scle fctor = length of lrge tringle (originl) 6 cm k = cm = Volume scle fctor = k k = k = volume of smll cone (imge) Volume scle fctor = volume of lrge cone (originl) 8 x cm = cm 5 6 Trnspose the eqution to get x = the unknown y itself. x = 60 Write your nswer. The volume of the smller cone is 60 cm. 8 We cn use the reltionship etween liner, re nd volume scle fctors to find ny unknown in ny pir of similr figures s long s scle fctor cn e estlished.. Given liner scle fctor (lsf) = k re scle fctor = k volume scle fctor = k For exmple: = = = = = 8. Given re scle fctor (sf) = k liner scle fctor = volume scle fctor = k For exmple: = k = = = = 8. Given volume scle fctor (vsf) = k liner scle fctor = re scle fctor = k For exmple: = 8 k = 8 = = = k k

51 Chpter 8 Geometry: similrity nd mensurtion 97 For two similr tringulr prisms with volumes of 6 m nd 8 m, find the totl surfce re of the lrger tringulr prism, if the smller prism hs totl surfce re of.5 m. volume of lrger prism (imge) Determine scle fctor, in Volume scle fctor = volume of smller prism (originl) this instnce the volume scle fctor, from the two known k 6 m = volumes. It is preferred tht 8 m the lrger unknown tringulr k = 8 prism is stted s the imge. Determine the re scle Liner scle fctor = k = k fctor. For ese of clcultion, k = 8 = chnge volume scle fctor to liner nd then to re scle Are scle fctor = k fctor. = = re of lrger prism (imge) Use the re scle fctor to Are scle fctor = re of smller prism (originl) find the totl surfce re. x m = m Trnspose the eqution to get unknown y itself. x =.5 x = 0 Write your nswer. The totl surfce re of the lrger tringulr prism is 0 m. 5 WORKED Exmple rememer rememer Are nd volume scle fctors The steps required to solve for length, re or volume using similrity re:. Clerly identify the known corresponding mesurements (length, re or volume) of the similr shpe.. Estlish scle fctor (liner, re or volume) using known pirs of mesurements.. Convert to n pproprite scle fctor to determine the unknown mesurement.. Use the scle fctor nd rtio to evlute the unknown. Are scle fctors re of imge Are scle rtio or fctor (sf) = re of originl = squre of liner scle fctor (lsf) = k Volume scle fctor volume of imge Volume scle rtio or fctor (vsf) = volume of originl = cue of liner scle fctor (lsf) = k

52 98 Further Mthemtics 8G Are nd volume scle fctors Mthcd Are nd volume scle fctors Complete the following tle of vlues. Liner scle fctors k Are scle fctors k Volume scle fctors k WORKED Exmple 9 Find the unknown re of the following pirs of similr figures. cm x mm 50 mm 5 mm.5 mm 8 cm 8 cm x cm c 7 m m.5 m x m d mm mm Surfce re Surfce re = x mm = 00 mm WORKED Exmple 0 Find the unknown length of the following pirs of similr figures. i ii 5 cm x m Are = 6.5 m.7 m Are =.0 m Are = 750 cm x Are = 000 cm

53 Chpter 8 Geometry: similrity nd mensurtion 99 c Two similr trpezium-shped strips of lnd hve n re of 0.5 hectres nd hectres. The lrger lock hs distnce of 50 metres etween the prllel sides. Find the sme length in the smller lock. Two photogrphs hve res of 8 cm nd 80 cm. The smller photo hs width of 6 cm. Find the width of the lrger photo. WORKED Exmple Find the unknown volume in the following pirs of similr ojects. Volume of smll pyrmid x cm = 0 cm 7 cm c 00 cm cm d cm cm 5 cm Volume of lrge sphere = 8 litres Volume = 00 cm 0 cm WORKED Exmple 5 For the similr tringulr pyrmids with volumes of 7 m nd m, find the totl surfce re of the lrger tringulr prism if the smller prism hs totl surfce re of.5 m. For sell with dimeter of 0 cm nd sketll with dimeter of 5 cm, find the totl surfce re of the sell if the sketll hs totl surfce re of 96.5 cm. c For inch cr tyre nd 0 inch truck tyre tht re similr, find the volume (to the nerest litre) of the truck tyre if the cr tyre hs volume of 70 litres. d For similr kitchen mixing owls with totl surfce res of 500 cm nd 75 cm, find the cpcity of the lrger owl if the smller owl hs cpcity of.5 litres (to the nerest qurter of litre). 6 Find the volume of the Find the volume of the lrger smll cone. tringulr pyrmid Are = 5 cm Are = 5 cm Volume of lrge cone = 70 cm TSA of smll pyrmid = 00 cm Volume of smll pyrmid = 000 cm TSA of lrge pyrmid = 88 cm

54 00 Further Mthemtics c Find totl surfce re of d Find the dimeter of the the smll prism smll cylinder. Are = cm cm TSA = 78 cm x cm Are = 6 cm TSA = Volume Volume x cm = 80 cm = 0 cm 7 A pln of holidy unglow hs scle of cm = 50 cm. Wht is the re of the pln? Express the drwing scle s liner scle fctor. c Using similrity, find the ctul re of the unglow (in m to deciml plces). d Wht is the re scle fctor (k )? cm 0 cm 5 cm 8 cm

55 Chpter 8 Geometry: similrity nd mensurtion 0 8 Wht is the re rtio of: two similr squres with side lengths of cm nd cm? two similr circles with dimeters of 9 m nd m? c two similr regulr pentgons with sides of 6 cm nd 0 cm? d two similr right-ngled tringles with ses of 7. mm nd.8 mm? 9 Find the volume rtios from the similr shpes given in question 8. TSA of lrge cone = 80 cm 0 Find the totl surfce re of the smll cone s given in the digrm. A : scle model of cr is creted from plster nd pinted. If the ctul cr hs volume of.5 m, find the mount of plster needed for the model to the nerest litre. The model needed 5 millilitres of pint. How much pint would e needed for the ctul cr (in litres to deciml plce)? Find the rtios of the volume of cues whose sides re in the rtio of :. An islnd in the Pcific Ocen hs n re of 500 km. Wht is the re of its representtion on mp drwn to scle of cm = 5 km? Two sttutes of fmous person used 500 cm nd.5 litres of cly. The smller sttue stood 5 cm tll. Wht is the height of the other sttue (to the nerest centimetre)? 5 The rtio of the volume of two cues is 7:8. Wht is the rtio of: the lengths of their edges? the totl surfce re? 6 The rdius of one sphere is equl to the dimeter of nother sphere. Find the rtio of the smll sphere to the lrge sphere: for totl surfce re for volume. 7 A cone is hlf-filled with ice-crem. Wht is the rtio of ice crem to empty spce? 8 multiple choice A :7 scle model of truck is mde from cly. The rtio of volume of the model to the rel truck is: A : B : C :9 D :79 E : multiple choice The rtio of the volume of the lue portion to the volume of the red portion is: A : B :8 C :9 D :6 E :7 0 multiple choice A :00 scle model of uilding is cue with sides of 00 cm. The volume of the rel uilding is: A m B m C m D m E 000 m h h

56 0 Further Mthemtics summry Properties of ngles, tringles nd polygons Drw creful digrms. Crefully interpret geometric nottions, such s the digrm t right. Crefully consider geometric rules, such s isosceles tringles hve equl sides nd ngles. Equl sides Are nd perimeter Perimeter is the distnce round closed figure. Circumference is the perimeter of circle. C = π rdius = πr Are is mesured in mm, cm, m, km nd hectres. cm = 0 mm 0 mm = 00 mm m = 00 cm 00 cm = cm km = 000 m 000 m = m hectre = m Are of shpes commonly encountered re:. Are of squre: A = l. Are of rectngle: A = l w. Are of prllelogrm: A = h. Are of trpezium: A = -- ( + ) h 5. Are of circle: A = πr 6. Are of tringle: A = -- h Are of composite figure = sum of the individul common figures A composite = A + A + A + A +... Totl surfce re (TSA) Totl surfce re (TSA) is mesured in mm, cm, m nd km. The TSAs of some common ojects re s follows:. Cues: TSA = 6l. Cuoids: TSA = (lw + lh + wh). Cylinders: TSA = πr(r + h). Cones: TSA = πr(r + s) where s is the slnt height 5. Spheres: TSA = πr For ll other ojects, form their nets nd estlish the totl surfce re formuls. Volume of prisms, pyrmids nd spheres Volume is the mount of spce occupied y -dimensionl oject. The units of volume re mm, cm (or cc) nd m mm = cm cm = m. litre = 000 cm. 000 litres = m

57 Chpter 8 Geometry: similrity nd mensurtion 0 Volume of prism, V prism = re of uniform cross-section height V = A H Volume of pyrmid, V pyrmid = re of cross-section t the se height V = A H The height of pyrmid, H, is sometimes cll the ltitude. Volume of sphere is V sphere = -- πr Volume of composite oject = sum of the individul common prisms, pyrmids or spheres. V composite = V + V + V +... or V composite = V V... Mps nd scles Rtio scle, for exmple :00, mens tht unit on the mp represents 00 units in rel life. SCALE : METRES KILOMETRES A simple conversion scle, for exmple cm = 00 m, mens cm on the mp represents 00 metres in rel life. Kilometres Kilometres

58 0 Further Mthemtics Similr figures C' Two ojects tht hve the sme shpe ut different size B' re sid to e similr. For figures to e similr, they must hve the following properties: 6 () The rtios of the corresponding sides must e equl. A' D' C' A B B C C D A D = common rtio B' = = = AB BC CD AD () The corresponding ngles re equl. A = A B = B C = C D = D 85 A' D' Scle fctor, k length of imge A B A B B C C A Scle fctor, k = = = = = length of originl AB AB BC CA where for enlrgements, k is greter thn nd for reductions, k is etween 0 nd. For k =, the figures re exctly the sme shpe nd size nd re referred to s congruent. A Similr tringles Two tringles re similr if one of the following conditions is identified:. All corresponding ngles re equl (AAA).. All corresponding pirs of sides re in the sme rtio (liner scle fctor) (SSS).. Two corresponding pirs of sides re in the sme rtio nd the included ngles re equl (SAS). Are nd volume scle fctors The steps required to solve for length, re or volume using similrity re:. Clerly identify the known corresponding mesurements (length, re or volume) of the similr shpes.. Estlish scle fctor (liner, re or volume) using known pirs of mesurements.. Convert to n pproprite scle fctor to determine the unknown mesurement.. Use the scle fctor nd rtio to evlute the unknown. Are scle fctor re of imge Are scle rtio or fctor (sf) = re of originl = squre of liner scle fctor (lsf) = k Volume scle fctor volume of imge Volume scle rtio or fctor (vsf) = volume of originl = cue of liner scle fctor (lsf) = k B A A C B B 85 A' D 5 60 B' D 9 9 C' C C

59 Chpter 8 Geometry: similrity nd mensurtion 05 CHAPTER review Multiple choice For the tringle shown in semicircle, x is: A B 58 C 68 D 90 E none of the ove A tringle LABC hs the following vlues given. AB = 0 cm, AC = cm where AB nd AC re perpendiculr. The re of the tringle is A 0 cm B 0 cm C 0 cm D cm E 60 cm The re of the kitchen ench shown in the pln is closest to: A 50π cm B 50π cm C 50π cm D 500π cm E 0 00 cm The totl surfce re of closed cylinder with rdius of 0 cm nd height of 0 cm is given y: A π 0 (0) B π 0 (0) C π 0 (00) D π 0 (60) E π 0 (60) 5 The net of n oject is shown in the digrm. An pproprite nme for the oject is: A rectngulr prism B rectngulr pyrmid C tringulr prism D tringulr pyrmid E trpezium prism 6 The volume of sphere with dimeter of 5 cm is closest to: A 560π cm B 900π cm C 500π cm D 500π cm E 6 000π cm 7 The volume of the composite oject, given tht VO = 0 cm is closest to: A 000 cm B 00 cm C 500 cm D 000 cm E cm 80 x 0 All mesurements in cm V O A 8B 8B 8C 8C 8D 8D

60 06 Further Mthemtics 8E 8 A mp rtio scle of : expressed s simple conversion scle is: A cm = 5 m B cm = 50 m C cm = 500 m D mm =.5 km E cm = 5 km 8E 8E 8F 8F 9 In the tringle shown, the vlue of c is: A B 6 C 9 D E 0 The circumference of the lrger cone is closest to: A mm B 5 mm C 6 mm D 0 mm E 59 mm The digonl distnce on the television screen is used to specify the different sizes ville. If the height on 5 cm television is 5 cm, then similr cm television hs height, h, which is closest to: A 67 cm B 5 cm C cm D 0 cm E 6 cm The digrm t right shows the pth of pool ll into the middle pocket of y 6 illird tle. To chieve this, the expression for the vlue of x is: A B C D E x = x 6 x = x x 6 = x 6 x = x + x = x 6 h cm 6 - x x c mm 5 cm mm 89 mm cm 5 cm 6