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GCSE RevisionMathematicsGeometryNew 9-1 GCSE Exam QuestionsGeometryGRADE 8/9 TARGETEasy print and cut. Answers Included.little-streams.comResource Ref No. MATF GEO GCSE8/9 - 2017 LittleStreams Ltd. All rights reserved.
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Teachers’ NoteGCSE Revision Geometry Grades 8/9What’s in this pack?This pack contains 48 exam style question cards covering the new curriculum for the top grades of thenew gcse in geometry. The design of the cards means you can use these in a variety of ways in yourclassroom; as a starter or a plenary, as extension work, as homework or as a revision tool. An answerkey is included.The questions feature some challenging questions, including topics new to the GCSE. This includesequation of a circle and circle theorems, proof with similar triangles, arcs, sectors and segments, volumeincluding cones and spheres, invariant points in transformations, englargement with area, advancedpolygon questions involving proof, trigonometry with sine and cosine rule and area of a triangle, specialtrigonometric values without a calculator, pythagoras in 3D, and vectors.Topics are as follows:Topic Card PageNumbersNumbersEquation of a Circle1-35Circle Theorems 4-9 5-7Congruence & Similarity10-11Arcs, Sectors & Segments 12-1677-8Volume17-20 9Invariant Points21-2410Enlargement 25-26 11Advanced Polygons 27-28 11Trigonometry 29-34 12-13Trigonometric Values 35-37 13-143D Pythagoras 38-41 14-15Vectors 42-48 15-16little-streams.com 2017 LittleStreams Ltd. All Rights Reserved.
1Grade 8/9 QuestionsEquation of a CircleNot Drawn to ScaleThe circle C has radius 3 and touches the y axis at (0,4).ya) Give the equation for the circle C.Equation of a Circleyb) P is a tangent to the circle. Findthe equation of the tangent at P.b) A line through the point P(5,-2) is atangent to the circle C at the point T.Find the length PT.5oP(3,4)xxMATF GEO GCSE8/9LittleStreams.co.ukGrade 8/9 QuestionsEquation of a CircleThe line l is a tangent to the circlex2 y2 100 at point A, and crossesthe x axis at P. Find the area ofthe triangle OAP.MATF GEO GCSE8/9Not Drawn to Scalea) Give the equation of the circle.432Grade 8/9 QuestionsNot Drawn to ScaleyA(8,6)oPxLittleStreams.co.ukMATF GEO GCSE8/94LittleStreams.co.ukGrade 8/9 QuestionsCircle TheoremsProve that the angle formedat the circumference when atriangle is drawn from bothends of a diameter is 90ºMATF GEO GCSE8/9LittleStreams.co.uk
5Grade 8/9 QuestionsNot Drawn to ScaleCircle TheoremsGiving reasons for youranswers, find the value of x.BNot Drawn to ScaleCircle TheoremsEF is a tangent that meets the circle at C. AC and BD arestright lines. Prove that X is not the centre of the circle.xA6Grade 8/9 QuestionsA54ºBo92ºXCFDD52ºCEMATF GEO GCSE8/97LittleStreams.co.ukGrade 8/9 QuestionsNot Drawn to ScaleCircle TheoremsAD and CD are tangents to the circle and ABE is astraight line. Find the size of angle CBE.o27º8LittleStreams.co.ukGrade 8/9 QuestionsNot Drawn to ScaleCircle TheoremsAA,B and C are points on a circle.BC bisects angle ABE. DBE is atangent. Angle CBE x.Prove that AC BC.ADMATF GEO GCSE8/9BCDEBCMATF GEO GCSE8/9xELittleStreams.co.ukMATF GEO GCSE8/9LittleStreams.co.uk
9Grade 8/9 QuestionsNot Drawn to ScaleCircle TheoremsDCABCDEF is a regular hexagon. D, V & Fare points on the circle. o is the centre Bof the circle. DC and FA are tangents tothe circle. Find the size of angle EFV.Give reasons for your answers.oMATF GEO GCSE8/9AGrade 8/9 Questions11 Congruence & Similaritya) Construct a line that is perpendicularto AC and passes through point B.Label where it crosses AC point D.b) Show that triangles ABD and BCD are similar.MATF GEO GCSE8/9Find the area of the shadedquadrilateral BCDEFLittleStreams.co.ukC10 Congruence & SimilarityNot Drawn to ScaleAB 4 5, AE 4BE, 4AD 5AEEVAGrade 8/9 QuestionsMATF GEO GCSE8/9CBDELittleStreams.co.ukGrade 8/9 Questions12 Arcs, Sectors & SegmentsNot Drawn to ScaleA7mBLittleStreams.co.ukABC is an arc of a circle with centreo40ºo and a radius of 7m. AC is achord of the circle. AOC 40º.7mCalculate the shaded area to 3 significant figures.Give your answer in cm2.MATF GEO GCSE8/9ABCLittleStreams.co.uk
Grade 8/9 Questions13 Arcs, Sectors & SegmentsNot Drawn to ScaleThe major sector shown has an areaof 189π cm2. Find the size of angle a.Grade 8/9 Questions14 Arcs, Sectors & SegmentsNot Drawn to ScaleSector ABC has centre C and DEF has centre F.D is the midpoint of AC.A18cma) By considering triangle DFC, calculatelength DC to 2 d. p.aD1.8cm160ºFBEC 60ºb) Calculate the perimeter of the shaded region ADEB to thenearest cm.LittleStreams.co.ukMATF GEO GCSE8/9Grade 8/9 Questions15 Arcs, Sectors & SegmentsNot Drawn to ScaleAThe circle has a radius of 5cm and arclength of 4.36 cm (to 2 d.p.).Find the shaded area (to 2 d.p.)MATF GEO GCSE8/9oBLittleStreams.co.ukMATF GEO GCSE8/9LittleStreams.co.ukGrade 8/9 Questions16 Arcs, Sectors & SegmentsA circle of radius 4cm is divided into equal sectors.The perimeter of each sector is 13.0265 (to 4 d.p.).How many sectors are there in total?MATF GEO GCSE8/9LittleStreams.co.uk
17Grade 8/9 QuestionsVolumeThe total volume for the solid is 128 8π.Find x without using a calculator.2x19 VolumeLittleStreams.co.ukGrade 8/9 QuestionsVolume of a sphere 3 πr34Sphere A has a radius of 3cm. The volume of sphereB is 50% greater than sphere A. What is the radius ofsphere B? Give your answer to 3 significant figures.MATF GEO GCSE8/9VolumeVolume of aVolume of aNot Drawn to Scale4sphere 3 31cone 3 2πrπr hA cone and a sphere have the same radius and thesame volume. What is the relationship between theheight (h) and the radius (r) of the cone?xxMATF GEO GCSE8/918Grade 8/9 QuestionsLittleStreams.co.ukMATF GEO GCSE8/920 VolumeLittleStreams.co.ukGrade 8/9 QuestionsVolume of a sphere 3 πr31Volume of a cone 3 πr2h4A bowl is a hemisphere with radius 4cm. Water fills threequarters of the volume of the bowl. The water is poured intoa hollow cone. The depth of the water in the cone is 12cm.Find the exact radius of the surface of the water in the cone.MATF GEO GCSE8/9LittleStreams.co.uk
Grade 8/9 Questions21 Invariant PointsThe square OABC has the following coordinates:O (0,0) A (0,2) B (2,2) C (2,0)Give coordinates of the invariant points of the square when.a) OABC is enlarged by scale factor 2 about (0,0)b) OABC is reflected in the line y 2LittleStreams.co.ukGrade 8/9 Questions23 Invariant PointsHow many invariant points are thereunder the following transformations?1098765432100 1 2 3 4 5 6 7 8 9 101098765432100 1 2 3 4 5 6 7 8 9 MATF GEO GCSE8/9Grade 8/9 Questions24 Invariant PointsGive the coordinates of the invariant pointsunder the following transformations.a) reflection along the line y xb) enlargment scale factor 2 centre (3,1)1098765432100 1 2 3 4 5 6 7 8 9 10c) rotation 360º about point (3,7)c) rotation 180º about point (5,2)MATF GEO GCSE8/9a) Under the transformation A, twovertices of the quadrilateral areinvariant. What is the transformation?ReflectionMATF GEO GCSE8/9b) reflection along the line y 522 Invariant Pointsb) Under the transformation B, four vertices are invariant.Which of these could be transformation B?c) OABC is reflected in the line y x.a) reflection along the line y xGrade 8/9 QuestionsLittleStreams.co.ukMATF GEO GCSE8/9LittleStreams.co.uk
25 EnglargementGrade 8/9 QuestionsCube A has a surface area of 2400cm2. Cube B isan enlargement of cube A by a scale factor of 14 .What is the volume of cube B?LittleStreams.co.ukMATF GEO GCSE8/9Grade 8/9 Questions27 Advanced PolygonsShape A is a regular polygon. The ratio of the sizeof the interior angle to the size of the exterior angleis 3:1. The ratio of the side length to the number ofsides is 3:2. Find the perimeter of shape A.MATF GEO GCSE8/9LittleStreams.co.uk26 EnlargementGrade 8/9 QuestionsTwo boxes are made with card. The boxes are similarcuboids. The smaller box has a height of 12cm.It takes 69% more card to make the larger box. Findthe height of the larger box.MATF GEO GCSE8/9LittleStreams.co.ukGrade 8/9 Questions28 Advanced PolygonsProve that the exterior angles of any polygon isequal to 360ºMATF GEO GCSE8/9LittleStreams.co.uk
29Grade 8/9 QuestionsNot Drawn to ScaleTrigonometryA7cm113ºin the shape ABCD, AB is 7cm Dand DC is 9cm. Angles ABC is9cm113º and BCD is 81º. Given that AC 11cm,find the area of ABCD to 2 decimal places.MATF GEO GCSE8/931 TrigonometryAFind the length of BCcorrect to 2 decimal places.B81ºCLittleStreams.co.ukGrade 8/9 QuestionsNot Drawn to Scale4.9cm62ºB30º80º5.7cmCDMATF GEO GCSE8/9LittleStreams.co.uk30 TrigonometryAB is 8.2 and DC is 9.2.Work out the length of BCto 2 decimal places.Grade 8/9 QuestionsNot Drawn to ScaleB8.2cmA45º70ºMATF GEO GCSE8/932 TrigonometryD9.2cmLittleStreams.co.ukGrade 8/9 QuestionsNot Drawn to ScaleBAB 2xAC 3xAAngle BAC 60ºShow that the perimeter of the triangle is (3 3)x.MATF ALG GCSE8/9CCLittleStreams.co.uk
33 TrigonometryGrade 8/9 QuestionsNot Drawn to ScaleThis shape has one line of symmetry.22mFind the value of angle acorrect to one decimal place.a34 TrigonometryGrade 8/9 QuestionsNot Drawn to ScaleB is a point on a circle. Find the heightof B above the horizontal ground.28mo20mB120ºA6mMATF GEO GCSE8/9LittleStreams.co.ukGrade 8/9 Questions35 Trigonometric ValuesFind the value of P without using a calculator:Sin(30º)4 PCos(30º) 12(MATF GEO GCSE8/9)LittleStreams.co.ukMATF GEO GCSE8/9LittleStreams.co.ukGrade 8/9 Questions36 Trigonometric ValuesWithout a calculator, show that 10sin(60)-9tan30 canbe written in the form a b where a and b are integers.MATF GEO GCSE8/9LittleStreams.co.uk
Grade 8/9 Questions37 Trigonometric ValuesWithout using a calculator, complete the followingtable:0ºSinCosGrade 8/9 Questions38 3D Pythagorasa) Find the exact length of AG.Not Drawn to ScaleHEATanGD42FB5Cb) Find the angle at which the diagonal AG meets thebase of the cuboid. Give your answer to 2 d.p.30º45º60º90ºMATF GEO GCSE8/9LittleStreams.co.ukGrade 8/9 Questions39 3D PythagorasENot Drawn to ScaleHGF5cmCHow much shorter (to 2d.p.) is theDA5cmdiagonal AG than the shortest9cmBdistance from A to G travelling along the cuboid’s edges?MATF GEO GCSE8/9LittleStreams.co.ukMATF GEO GCSE8/9LittleStreams.co.ukGrade 8/9 Questions40 3D PythagorasThe diagram shows a regular pyramidwith apex V. Calculate the exactFAheight of V above the base of3cmthe pyramid.BMATF GEO GCSE8/9Not Drawn to ScaleV7cmE3cmCD3cmLittleStreams.co.uk
Grade 8/9 Questions41 3D PythagorasNot Drawn to ScaleAWithout using a calculator, findthe length of the diagonal AB.120cm30cm B40cmGrade 8/9 Questions42 VectorsNot Drawn to ScaleIn triangle ABC, M is the midpoint of ACand N is a point on BC where NB:NC 1:3.AC 2aAB 5bBNa) Work out MN in terms of a and b inits simplest form.ACMb) Is MN parallel to AB? Give reasons.MATF GEO GCSE8/943LittleStreams.co.ukGrade 8/9 QuestionsVectorsO3aOA 3a and OB 2b. M is the midpoint of AB.a) Find OM in terms of a & b.Not Drawn to ScaleAM2bBMATF GEO GCSE8/944LittleStreams.co.ukGrade 8/9 QuestionsNot Drawn to ScaleVectorsABCD is a parallelogram.AB 2a and BW b.M is the midpoint of CDand AX 2XC. BW:BC 1:6Bb WCX2aAMDFind BX in terms of a and b in its simplest form.b) X is a point on AB and AX:XB 1:3. Find OX in terms of a & b.MATF GEO GCSE8/9LittleStreams.co.ukMATF GEO GCSE8/9LittleStreams.co.uk
45Grade 8/9 QuestionsNot Drawn to ScaleVectorskbB5aMATF GEO GCSE8/9AC DGrade 8/9 QuestionsM is the midpoint of BC.Express EM in terms of a and b in itssimplest form.Not Drawn to ScaleAB10aF10bCoE(-3-1 )(-1-5 )AB Find the exact value of cos(ACB) in its simplest form.LittleStreams.co.ukVectors46 VectorsTwo vectors are defined as follows:ABCD is a parallelogram,AB 5a and AD kb.AWhen the line AC is extended,E is a point on AC such that BE 3a 16b.Calculate the value of k.47CGrade 8/9 QuestionsDMATF GEO GCSE8/948 VectorsLittleStreams.co.ukGrade 8/9 QuestionsA12aABCO is a parallelogram. P is thepoint on AC such that AP:PC 2:1. Oa) Find the vector OP in terms of a and b.Not Drawn to Scale12bBPCb) M is midpoint of CB. Prove that OPM is a straight line.MATF GEO GCSE8/9LittleStreams.co.ukMATF GEO GCSE8/9LittleStreams.co.uk
AnswersGCSE Revision: GeometryGrade 8 and 9 Target - Page 1 of 21a) (x 3)2 (y-4)2 9b) Sketch the information. (-3-5)2 (4--2)2 1032 PT2 102 PT 913Grad of OA 6/8 so grad of AP -8/6Equation AP (use 8,6) is 6 -4/3(8) cc 50/3 so y -4/3x 50/3. When y 0, x 12.512.5 x 6 / 2 37.5 so area 37.55Angle DBC 54º (angles in the samesegment are equal).Angle ABC 90º (angle in semi-circle is 90º).Angle x 90º - 54º 36º.7CDO ADO 27. DAO and DCO 90º27 27 54. 180-54 126 so AOC 126ABC 126 2 63.CBE 180-63 117.9DEF 180-(360 6) 120º FEV 180-120 60ºCDo & AFo 90º as tangent to circleperpendicular to radius. DoF 720-3(120)-2(90) 180º. DVF 90º as angle at ceture is twice angleat circumference. EFV 180-90-60 30º3 T1011a) Construction marks need to be seen.b) ADB CDB 90º. DCB CBD 90ºABD DAB 90º. ABD CBD 90ºDCB ABD and DAB CBDso triangles are similar.13189π θ/360 x 182 x πθ 189π x 360 (182π)θ 210360 - 210 150 so a 15015θ/180 x 5π 4.36. θ 49.96.49.96/360 x 52π 10.9cm2Area triangle (0.5)(5)(5)sin49.96 9.5702cm210.9-9.5702 1.32978. 1.33cm217Cuboid volume 2x3Half cylinder (1/2x)2(π)(x)/2 1/8x3πTotal Volume 2x3 1/8x3πCompare Coefficients: 2x3 128 x 419Volume of sphere A 4/3πx33 36π cm3.Volume of sphere B 36π 18π 54πr3 54π 4/3π 40.5r 3.434142728. 3.43cm21a) (0,0)b) (0,2) (2,2)c) (0,0) (2,2)23a) No invariant points.b) 2 invariant points.c) 1 invariant point.P24a) x2 y2 25b) Grad op 4/3 so grad tangent -3/44 -3/4(3) c c 25/4y -3/4x 25/4 4y -3x 25aabba a b b 180º (angles in triangleadd to 180º)2a 2b 180ºa b 90ºSo the angle is a right angle.6By alternate segment theorem, angle CAD 52º.DXC 180-92 88.If X was at the centre of the circle then angleDXC 2 times CAD 104º 88º, so X is not centre.8CBE x and ABC x.Alternate angle theorem CAB x.Two equal angles means triangle isisosceles so AC BC.10Let BE x & AE 4x. Pythag x2 4x2 8012Sector area (40/360)(72)π (49/9)πTriangle area (0.5)(7)(7)sin(40) 15.7482.(49/9)π-15.7482. 1.35593.m2 135559.3cm2 13559cm214a) DFC (360-160) 2 100.DC sin(100) x (1.8 sin(30)) 3.55cmb) AC 2 x DC 7.09 Arc DE 1.8π x 160/180 8/5π.Arc AB 60/180 x 7.09π 7.4252.7.4252. 8/5π 3.55 3.55 19.551 20 cm1613.0265-4-4 5.0265Circumference 8π 25.13274.25.13274. 5.0265 518Area Sphere 4/3πr3, Area Cone 1/3πr2h4/3r3 1/3r2h4r3 r2hh 4r20Volume of water (4/3πr3) 2 x (3/4) 32π1/3πr2 x 12 32πr2 8r 2 222a) Reflection in the line x yb) Translation (Vector 0,0)Rotation (360º)Enlargement (Scale Factor 0)24a) (2,2) (3,3) (8,8)b) (3,1)c) (3,1) (2,2) (3,7) (8,8) (3,3)Similar triangles: AE 8 so AD 8x5/4 10, ED 2BE 4 so CD 4x5/4 5Area of Trapezium (5 4)/2x2 9 units2little-streams.com - 2017 LittleStreams Ltd. - MATF GEO GCSE8/9
AnswersGCSE Revision: GeometryGrade 8 and 9 Target - Page 2 of 225Scale factor are pf 1/4 means multiply by(1/4)2 1/16. 2400 x 1/16 150.Cube B has surface area 150. 150 6 25 25 55x5x5 125cm32669% means 169% larger. 1.69 1.31.3 x 12 15.6Alternatively make it a cuboid 12x12x1227Int:Ext 3:1 Int Ext 180 180 4 45Exterior angles 45º, 360 45 8 The shapehas 8 sides.Length:Sides 3:2 12:8. 12x8 96.28Interior Exterior 180º Let sum of exteriorangles be E, and interior angles be I.E I 180n E 180n-II 180(n-2)E 180n-180n (180x2) E 360.29Break the quadrilaterals into two triangles.Triangle ABC: sin(113)/11 sin(BCA)/7 BCA 35.86o81-35.86 45.14Area of triangle ABC 0.5x7x11xsin(180-113-35.86) 19.910.Area of triangle ADC 0.5x11x9xsin(45.14) 35.087.19.910. 35.087. 54.99cm230DB/Sin(45) 8.2/Sin(70) 6.1703.(BC)2 6.17032 9.22-2x6.1703x9.2x cos(110)BC 12.69232611BC 12.6931BD2 4.92 5.72-2x4.9x5.7xcos(62) 5.5023.Angle BDC 180-120 605.5023./sin(80) BC/sin(60)BC 4.838642471. BC 4.8432Cosine rule: a2 (x)2 (2x)2-(2)(x)(2x)Cos60ºa2 5x2-(4x2)(1/2), a2 5x2-2x2, a2 3x2, a 3xx 2x 3x 3x 3x (3 3)x33(22-6) 2 8tan(a) 28/8a tan-1(28/8) 74.0546041.a 74.13435Sin(30º) 12 , Cos(30º) 32 , 12 32 3124 264( 31 2 3)2 ( 2 3 2 3) (2 3)2 3612 3p 3370º30ºSinCosTan1/2 3/21/ 3010SinCos45º1/ 21/ 290º1060º 3/21/2Tan1 3undefined39AG2 AB2 AC2 CG2 92 52 52 131.AG 11.445523.9 5 5 19.19-11.445523. 7.5544 7.5541AB2 302 402 1202 900 1600 14400AB2 16900AB 130cm43a) OM OA 1/2AB OA 1/2(-3a 2b) 3/2a b.b) OX OA AX. As AX:XB 1:3, AX 1/4 of AB.So OX OA 1/4AB 3a 1/4(-3a 2b)OX 3a-3/4a 3/4b 9/4a 3/4b.45ACE is a straight line so AC and AE are parallel vectors.AC kb 5a. AE AB BE 5a 3a 16b 8a 16bCompare Coefficent of a. AE 8/5 ACk 16 8/5 1047EX AB OB 1/2 OA -AO OB OB 1/2 OA -10a 10b 10b 5a -5a 20bo20m30º120ºBOP120-90 30º20sin(30) OP 1020 10 30A36Sin(60) 3/2 tan(30) 3/310 3/2-9 3/3 5 3-3 3 2 338a) AC2 42 52 41 AC 41AG2 412 22 45 AG 3 5b) Sinx 2/3 5x Sin-1(2/3 5)x 17.3540Base made of equilateral triangles.h2 72-32h2 40h 2 1042a) MC a CB CA AB -2a 5bCN 3/4(-2a 5b) MN MC CN a 3/4(-2a 5b)3/4a-6/4a 15/4b -3/4a 15/4bb) MN is not multiple of AB so lines not parallel44BX BC CX, BC 6BW 6b, AX 2XCCX 1/3CA -6b-2a 1/3(-6b-2a) -2b-2/3aBX 6b-2b-2/3a 4b-2/3a46487 h3-222BC BA AC ( 15 ) (-3-1) ( 4 ) AC (-3) (-1) 102222AB (-1) (-5) 26, BC (-2) (4) 20 2 510 20-26Cosine Rule: CosA 2x 10x2 5 420 2a) AC -12a 12b. AP 2/3(12a 12b) -8a 8b.OP 12a-8a 8b 4a 8bb) PM PC CM 1/3(-12a 12b) 1/2(12a) 2a 4bScalar multiple sharing point P, so O, P, M are co-linear.little-streams.com - 2017 LittleStreams Ltd. - MATF GEO GCSE8/9
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Circle Theorems Circle Theorems Circle Theorems Circle Theorems A,B and C are points on a circle. BC bisects angle ABE. DBE is a tangent. Angle CBE x. Prove that AC BC. Grade 8/9 Questions Grade 8/9 Questions Grade 8/9 Questions Grade 8/9 Questions LittleStreams.co.uk LittleStreams.co.uk LittleStreams.co.uk MATF_GEO_GCSE8/9 MATF_GEO_GCSE8/9 .
