Euclidean Geometry & Measurement

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1 Euclidean Geometry & Measuerement Worksheet Compiled by : T Remias 2015 Page 1

2 WORKSHEET ON EUCLIDIAN GOEMETRY QUESTION Complete the statements below by filling in the missing word(s) so that the statements are CORRECT The angle subtended by a chord at the centre of a circle is.... (1) The angle between the tangent and a chord is... (1) The opposite angles of a cyclic quadrilateral are... (1) 1.2 In the figure below, RDS is a tangent to a circle O at D. If BC = DC and. Calculate with reasons the measures of : (2) (1) (1) (2) [09] Page 2

3 QUESTION 2 In the diagram below, points R, P, A, Q and T lie on a circle. RA bisects and AB = AQ. RA and TQ produced meet at B Prove that : 2.1 AQ bisects (3) 2.2 TR = TB (2) 2.3 (3) [08] Page 3

4 QUESTION 3 In the figure below, PQ is a diameter to circle PWRQ. SP is a tangent to the circle P. Let 3.1 Why is P 3.1 Prove that (3) (1) 3.3 Prove that SRWT is a cyclic quadrilateral. (3) 3.4 Prove that WR /// ST (3) 3.5 If QW = 5cm, TW= 3cm, QR= 4cm and WR= 2cm, calculate the length of: TS (3) SR (3) [16] Page 4

5 QUESTION 4 In the figure below, ABC has D and E on BC. BD 6cm and DC cm. AT: TC 2: 1 and AD ǁ TE. 4.1 Write down the numerical value of (1) 4.2 Show that D is the midpoint of BE. (2) 4.3 If FD = 2cm, calculate the length of TE. (2) 4.4 Calculate the numerical value of: (1) (3) [9] Page 5

6 QUESTION Complete the statement below by filing in the missing word(s) so that the statement is CORRECT: The angle subtended by a chord or arc at the centre of a circle is... (1) 5.2 In the figure below, O is the centre of the circle and PT = PR. Let Express in terms of. (3) If TQ = TR and 12, calculate the measure of: (a) (2) (b) (Hint: Draw QR) (3) [9] Page 6

7 QUESTION 6 In the figure TP and TS are tangents to the given circle. R is a point on the circumference. Q is a point on PR such that. SQ is drawn. Let Prove that: 6.1 T ǁ SR (4) 6.2 QPTS is a cyclic quadrilateral (4) 6.3 TQ bisects S P (3) [11] Page 7

8 QUESTION 7 In the figure A ǁ RT, 7.1 If BT =, calculate TQ in terms of (3) 7.2 Hence, or otherwise, calculate the numerical value of: (3) (4) QUESTION 8 In the accompanying figure, AB is the diameter of circle ADCB. Chords AC and BD intersect at E. EP is perpendicular to AB. [10] 8.1 Prove that BPE ǁǁ BDA (3) 8.2 Hence show that (2) 8.3 Prove that (5) [10] Page 8

9 QUESTION In the diagram below, O is the centre of the circle. K, L and M are points on the circumference of the circle. Prove that the obtuse angle at O, (6) 9.2 In the diagram below, O is the centre of the circle. P, Q, R and S are a point on the circumference of the circle. TOQ is a straight line such that T lies on PS. PQ = QR and Calculate, with reasons. in terms of. (3) Show that TQ bisects. (3) Show that STOR is a cyclic quadrilateral (3) [15] Page 9

10 QUESTION 10 O is the centre of the circle below. OM AC. The radius of the circle is equal to 5cm and BC = 8cm 10.1 Write down the size of (1) 10.2 Calculate: QUESTION The length of AM, with reasons (3) Area AOM: ABC (3) [7] In the figure below, GB ǁ FC and BE ǁ CD. AC 6cm and 11.1 Calculate with reasons: AH: ED (4) Page 10

11 (2) 11.2 If HE = 2cm, calculate the value of AD HE (2) [8] QUESTION 12 In the figure below, AB is a tangent to the circle with centre O. AC = AO and BA ǁ CE. DC produced, cuts tangent BA at B Show (3) 12.2 Prove that ACF ADC. (3) 12.3 Prove that AD = 4AF (4) [10] Page 11

12 QUESTION Complete the statement The sum of the angles around a point is... (1) 13.2 In the figure below. O is the centre of the circle. K, L, M and N are points on the circumference of the circle such that LM = MN. Determine with reasons, the values of the following: (3) (3) [6] Page 12

13 QUESTION Complete the following statement: The angle between the tangent and the chord... (1) 14.2 In the diagram below, two circles have common tangent TAB. PT is a tangent to the smaller circle. PAQ, QRT and NAR are straight lines. Let Name, with reasons, THREE other angles equal to. (5) Prove that APTR is a cyclic quadrilateral. (5) [10] Page 13

14 QUESTION 15 Two circles touch each other at point A. The smaller circle passes through O, the centre of the larger circle. Point E is on the circumference of the smaller circle. A, D, B and C are points on the circumference of the larger circle. OE CA Prove, with reasons that AE = BE (2) 15.2 Prove that AED CEB (3) 15.3 Hence, or otherwise, show that (2) 15.4 If AE.EB = EF.EC, show that E is the midpoint of DF (3) [10] Page 14

15 QUESTION 16 ABC is a right-angled triangle with AB such that DE AB. E and D are joined.. D is a point on AC such that BD AC and E is a point on AD: DC = 3: 2 AD = 15cm 16.1 Prove that BDA CDB. (3) 16.2 Calculate BD (Leave your answer in surd form). (3) 16.3 Calculate E (Leave your answer in surd form). (6) [12] Page 15

16 QUESTION 17 In the diagram below AC is a diameter of the circle O. AC and chord BD intersect at E. AB, BC and AD are also chords of the circle. OD is joined. AE BD. If, calculate, with reasons, the size of: 17.1 (3) 17.2 (2) 17.3 Show that AE bisects (3) [8] QUESTION In the diagram below O is the centre of the circle. GH is a tangent to the circle at T.J and K are points on the circumference of the circle. TJ, TK and JK are joined. Prove the theorem that states. (5) Page 16

17 18.2 ED is a diameter of the circle, with centre O. ED is extended to C. CA is a tangent to the circle at B. AO intersects BE at F. BD AO Write down, with reasons, THREE other angles equal to (4) Determine, with reasons, (3) Prove that F is the midpoint of BE. (4) Prove that CBD CEB (2) Prove that 2EF.CB = CE.BD (3) [16] Page 17

18 QUESTION 19 In the diagram below A, B, C and D are points on the circumference of the circle. BD and AC intersect at E. Also, EB = 8cm, AE: EC = 4:7 If and Calculate [6] QUESTION 20 In the diagram below M is the centre of the circle. FEC is a tangent to the circle at E. D is the midpoint of AB Prove MDCE is a cyclic quadrilateral. (3) 20.2 Prove that (3) 20.3 Calculate CE if AB = 60mm, ME = 40mm and BC = 20mm (4) [10] Page 18

19 QUESTION 21 O is the centre of the circle. AB produced and DO produced meet at C. BC = OA and. Calculate with reasons, QUESTION 22 [5] 22.1 In the figures below O is the centre of the circle and PRST is a cyclic quadrilateral. Prove the theorem that states [5] Page 19

20 QUESTION 23 In the figures below DE FG BC. AD = 36cm, DF = 24cm, AE = 48cm and DE = GC =40cm. Determine with reasons, the length of: 23.1 EG (2) 23.2 BC (4) [6] QUESTION 24 In the diagram below DA is a tangent to the circle ACBT at A. CT and AD are produced to meet at P. BT is a produced to cut PA at D. AC, CB, AB and AT are joined. AC BD. Let 24.1 Prove that ABC ADT. (6) 24.2 Prove that PT is a tangent to the circle ADT at T. (3) Page 20

21 24.3 Prove that APT TPD. (3) 24.4 If show that (4) QUESTION In the diagram below, O is the centre of the circle, PQ is a tangent to the circle at A. B and C are points on the circumference of the circle. AB, AC and BC are joined. [16] Prove the theorem that states (6) 25.2 RS is a diameter of the circle with centre O. Chord ST is produced to W. Chord SP produced meets the tangent RW at V. Calculate the size of (1) (2) (3) Prove that (4) [16] Page 21

22 QUESTION 26 AB is a diameter of the circle ABCD. OD is drawn parallel to BC and meets AC in E. If the radius is 10cm and AC = 16cm, calculate the length ED. [5] QUESTION 27 CD is a tangent to circle ABDEF at D. Chord AB is produced at C. Chord BE cuts chord AD in H and chord FD in G. AC FD and FE = AB. Let = y 27.1 Determine THREE other angles that are each equal to. (6) 27.2 Prove that BHD FED. (5) 27.3 Hence, or otherwise, prove that AB.BD = FD.BH (2) [13] Page 22

23 QUESTION 28 In the diagram below, AM is the diameter of the bigger circle AMP. RPS is a common tangent to both circles at P. APB and MPN are straight lines State the size of 1 (1) 28.2 Hence show that BN is the diameter of the smaller circle. (2) 28.3 If 1, calculate the size of each of the following angles : (1) (1) (2) [7] QUESTION 29 In the diagram below, O is the centre of the circle with diameter RK. PS K K intersect at 29.1 PS = 4x, write down the length of ST in terms of x (1) Page 23

24 29.2 Prove that RST PKT. (3) 29.3 If it is further given that TK = x and RT = 320mm, calculate the value of x (3) [7] QUESTION 30 In P W, S is a point on W such that SR ǁ P. T is a point on W such that ST ǁ PR. RT = 6cm WS: SP = 3: 2 Calculate: 30.1 WT (3) 30.2 WQ (4) [7] QUESTION In the diagram below, O is the centre of the circle. PSRT is a cyclic quadrilateral. Prove the theorem that states (5) Page 24

25 31.2 In the diagram below, O is the centre of the circle. AB is a diameter of the circle. Chord CF produced meets chord EB produced at D. Chord EC is parallel to chord BF. CO and AC are joined. Let QUESTION Determine in terms of the size of. (4) Prove that DF = BD. (4) Show that. (4) IF DF 5cm and OA 6cm, calculate area BFD: area AOC. (4) [22] AB and CD are two chords of a circle with centre O. M is on AB and N is on CD such that OM AB and ON CD. Also AB = 50mm, OM =40mm and ON = 20mm. Determine the radius of the circle and the length of CD [4] Page 25

26 QUESTION 33 O is the centre of the circle below and Determine 2 and in terms of x (4) 33.2 Determine 1 and 2 in terms of x (4) 33.3 Determine 1 +. What do you notice (2) 33.4 Write down your observation regarding the measures of 2 and. (2) [12] QUESTION 34 O is the centre of the circle above and MPT is a tangent. Also, OP MT. Determine, with reasons the value of x, y and z. [5] Page 26

27 QUESTION 35 Given : AB = AC, AP//BC and 2 = 2 Prove that : 35.1 PAL is a tangent to the circle ABC (6) 35.2 AB is a tangent to circle ADP (4) [10] QUESTION 36 In the accompanying figure, two circles intersect at F and D BFT is a tangent to the smaller circle at F. Straight line AFE is drawn such that FD = FE. CDE is a straight line and chord AC and BF cut at K Prove that : 36.1 BT//CE (3) 36.2 BCEF is a parallelogram (5) 36.3 AC = BF (4) [12] Page 27

28 QUESTION 37 D is the midpoint of the chord AB and DC AB with C on the circle. If AB = 300mm, and CD = 50mm, calculate the radius of the circle. QUESTION 38 AB is the chord of the circle with centre O and is 24cm long. C is the midpoint of AB. CE AB cuts the circle at E. Calculate the value of x if CE = 8cm (AC =...cm) [5] [5] Page 28

29 QUESTION 39 Given O is the centre of the circle, determine the unknown angle in each of the following diagrams Page 29

30 QUESTION 40 Find the values of the unknown angles Page 30

31 40.3 QUESTION 41 Prove that ABCD is a cyclic quadrilateral Page 31

32 QUESTION 42 Determine the unknown letters, stating reasons Page 32

33 QUESTION 43 D, E, F and G are points on circle with centre M. 1 = 7 0 and 2 = Determine the sizes of the following angles, stating reasons: (3) (2) (1) 43.4 (2) (1) [9] Page 33

34 QUESTION 44 DF is a tangent to the circle at E. K F = 35 0, O G = 31 0 and EK = HK. O is the centre of the circle Determine : (3) 44.2 (3) 44.3 D H (3) 44.4 (2) (3) 44.6 O D (1) (2) 44.8 What type of quadrilateral is EOHK? Justify your answer. (3) 44.9 Prove that OK perpendicularly bisects EH. (3) If KO is produced, will it pass through G? Justify your answer. (3) [26] Page 34

35 QUESTION O is the centre of the circle below. P, Q, R, S and T lie on the circumference of the circle and OR QS Prove that OQRS is a kite. (3) Determine the size of a, b, c, d, e, f and g (7) If QS = 48 units and VR = 2units, determine the length of OQ, the radius of the circle (3) Page 35

36 45.2 The chord QR produced, meets tangent US at S. The centre of the circle is at O. SU touches the circle at T. P U = 31 0 and R T = Determine the size of : Q T (3) Q T (3) Q U (3) Determine the angles a, b, c, d, e, f, g, h, i, j, k, l, m, n, w, x, y and z in that order (18) [40] Page 36

37 QUESTION 46 O is the centre of the circle. The diameter AG is produced to C. EC is a tangent to the circle at D. A, D, G and F lie on the circle and DC = BC. AFB is a straight line and 1 = x 46.1 Prove that ADCB is a cyclic quadrilateral. (4) 46.2 Prove that AC bisects B D (2) 46.3 Prove that DG = FG (2) 46.4 Determine with reasons, the size of 2 (3) 46.5 Prove that 1 = 2 (6) 46.6 Prove that FBCG is a cyclic quadrilateral (3) 46.7 For what value of x will DC be a tangent to the circle through F, B, C and G (3) [23] Page 37

38 QUESTION Complete the statement so that it is valid : The line drawn from the centre of the circle perpendicular to the chord... (1) 47.2 In the diagram below, O is the centre of the circle. The diameter DE is perpendicular to the chord PQ at C. DE = 20cm and CE = 2cm Calculate the length of the following with reasons : OC (2) PQ (4) [7] Page 38

39 QUESTION 48 In the diagram, M is the centre of the circle. A, B, C, K and T lie on the circle. AT produced and CK produced meet in N. Also NA = NC and 48.1 Calculate, with reasons, the size of the following angles: K A (2) (2) (2) (2) 48.2 Show that NK = NT (2) 48.3 Prove that AMKN is a cyclic quadrilateral (3) [13] Page 39

40 QUESTION Complete the following statement so that it is valid : The angle between a chord and a tangent at the point of contact is... (1) 49.2 In the diagram below, EA is a tangent to circle ABCD at A. AC is a tangent to the circle CDFG at C. CE and AG intersect in D If 1 = x and 1 = y, prove the following with reasons : BCG AE (5) AE is a tangent to circle FED (5) AB = AC (4) [15] Page 40

41 QUESTION 50 In the diagram below AC and BC are tangents to the circle. A, B, P and Q lie on the circle. A C = 30 0 and O is the centre of the circle Prove that AOBC is a kite (4) 50.2 Prove that AOBC is a cyclic quadrilateral (4) 50.3 Determine the size of P Q, 2, A B and A B (5) 50.4 Prove that APBC is a rhombus. (4) [17] Page 41

42 QUESTION 51 In the figure below, O is the centre of the circle and DB = DF. AF, BE and BF are straight lines and C D = Find, stating reasons the magnitude of the following angles : (3) 51.2 (3) (2) (2) (2) [12] Page 42

43 QUESTION 52 ABCD is a cyclic quadrilateral with BC = CD. ECF is a tangent to the circle at C. AB and AD produced meet ECF at E and F. Prove that : 52.1 EF BD (3) 52.2 (5) 52.3 (4) [12] QUESTION 53 Consider a right angled triangle ABC with. Let BC = a and AB = c. let D be on AC such that BD etermine the length of in terms of a and c [4] Page 43

44 QUESTION Triangle ABC has B = 90. Point D is on AC such that BD AC. Let BC = a and AC = c Prove that ABC BDC. (4) Hence show that BC 2 = DC.AC (1) Determine the length of BD in terms of a and c. (4) ABCDE has BE CD. AB = 3 units, BC = 2 units and AE = 4,5 units Calculate the length of AD. (3) If CD = 8 units, calculate the length of BE. (3) [15] Page 44

45 QUESTION Triangle ABC has F as the midpoint of BC. D is a point on AB. CD and AF intersect at G. E is a point on BC such that AB GE. AG : GF = 1 : 2 Calculate the numerical value of the following ratios, giving reasons: (4) (4) ABCE is a cyclic quadrilateral with AE and BC produced to D, such that AB = CD Prove that CDE ADB (4) Hence show that CD 2 = CE.AD (3) [15] Page 45

46 QUESTION Cyclic quadrilateral ABCD has AD produced to F and AB produced to E. EF BC. E 2 = x Name, with reasons, two other angles equal to x. (3) Prove that DEF EAF (4) Hence show that EF 2 = DF.AF (1) ABC has perpendicular bisector CD of side AB. F is a point on BC such that FC: BC = 2 : 5.E is a point on AB such that CD FE Calculate the numerical value of the ratio. (4) Calculate the numerical value of the ratio. (3) [15] Page 46

47 QUESTION MP is a tangent to circle MNO at M. NO is produced to P. Points R and S are on MN and NO respectively such that MO RS. Q is a point on MP such that OQ MN Prove that PMO PNM (4) Hence, calculate the length of NO if PM = 6 cm and PO = 3,6 cm. (4) Determine the numerical value of, given NS = 2 cm. (5) ABC has points D on AC and E on BC such that BDE = 90. AB = 4 units, BD = 4 units, DE = 3 units and CE = 2 units. Calculate the length of AC. (2) [15] Page 47

48 QUESTION ABC has points F and E on AB and AC respectively such that FE BC and points D and G on AC and BC respectively such that DG AB. In addition, AF = 3 cm, FB = 9 cm and AE = 2 cm. GC : BG = 1 : Calculate, giving reasons, the length of EC. (5) Calculate, giving reasons, the length of ED. (5) ABD has point C on BD such that AC BD. BC = 10 cm and CD = 20 cm. Calculate AC, correct to two decimal places. (5) [15] Page 48

49 QUESTION ABC has points D and E on AB and points F and G on BC such that AC DG EF Calculate DE. (3) Calculate AC. (4) Cyclic quadrilateral PQRS has PQ = PS. PT is a tangent to the cyclic quadrilateral at P Let P 1 = x. Determine, with reasons, two other angles equal to x. (2) Show that TP 2 = QT.RP (6) [15] Page 49

50 QUESTION Circle ABC has centre O and diameter COA produced to T. SBT is a tangent to the circle at B. SO intersects BC at R such that BC SO Let B 1 = x. Determine, with reasons, three other angles equal to x. (5) Determine TBC and TAB in terms of x. (2) Prove that TBA TCB (4) ABC has points D and E on BC and AC respectively such that AB ED. BD = 15 cm, DC = 5 cm and EC = 3 cm Calculate the length of AE. (2) Calculate the length of AB. (2) [15] Page 50

51 QUESTION Two circles intersect at C and D. ABCD is a cyclic quadrilateral in the larger circle and CDEF is a cyclic quadrilateral in the smaller circle. BED and AEF are straight lines. Let C = x Determine, giving reasons, two other angles equal to x. (2) Write, giving reasons, BAD and E 3 in terms of x. (2) Prove that BA 2 = BD.BE (6) ABC has DE joining the midpoints of AB and BC. CE = 4 units and DB = 3 units Show that DC 2 = DE 2 + CE CE.EB (3) Hence calculate the length of DC correct to one decimal place. (2) [15] Page 51

52 QUESTION Two circles intersect at A and D. The larger circle has cyclic quadrilateral ABCD with BD extended to T. TSA is a tangent to the larger circle. CD is extended to point S on the smaller circle. Let A 3 = x Determine, with reasons, five other angles equal to x. (5) Prove ACB ASD (4) Prove TD is a tangent to the circle ADS. (2) ABC has points D and E on AC and BC respectively such that AB DE. AC = 20 units. DC = 14 units and BE = 8 units Calculate the length of EC. (2) Calculate the length of AB, correct to one decimal place, for ABC to be a rightangled triangle with A = 90. (2) [15] Page 52

53 QUESTION ABC has points D and F on AB such that CD EF and D is the midpoint of AB. G is a point on CD and E is a point on BC such that GE AB. DG : GC = 1 : Calculate the numerical value of the ratio (3) Prove GE = AB (4) Two circles with centres B and F intersect at A and D. CA is a tangent to circle F at A. EA is a tangent to circle B at A. CD is a chord on circle B and DE is a chord on circle F Prove, giving reasons, that ADE CDA (4) Show, giving reasons, that B 1 = F 1 (4) [15] Page 53

54 QUESTION 64 In the figure below AC is a tangent to the circle at B. O is the centre of the circle. BC is perpendicular to DC and 1 = x. Prove that : 64.1 DC KR (4) = (2) 64.3 BCRO is a cyclic quadrilateral (4) 64.4 CR is a tangent to the circle through E, K and O (3) 64.5 S is the centre of the inscribed circle of (incentre) (5) [18] Page 54

55 QUESTION 65 In the sketch below TA and TB are tangents to the circle with centre O. TQP is a secant and chord AC is parallel to QP. QP cuts AB and BC in H and K respectively. Prove that: 65.1 AOBT is a cyclic quadrilateral (3) 65.2 T B = T B (3) 65.3 AKBT is a cyclic quadrilateral (3) 65.4 TA is a tangent to the circle AHK (3) [12] Page 55

56 QUESTION 66 In the diagram below DB is a tangent to the left hand circle at B. CA bisects B E and AC is a straight line. DB. DAE 66.1 If 2 = x, list, giving reasons, seven other angles each equal to x (10) 66.2 Prove that CA is a tangent to the circle ABD (4) 66.3 If EA = EC, prove that BE and BA divide D C into three equal parts (4) [18] Page 56

57 WORKSHEET ON AREA AND VOLUME QUESTION 1 1 Consider the figures below and in each case determine: 1.1 the surface area 1.2 the volume (a) (b) 15 cm 24 cm 30 cm 18 cm 18 cm 1.3 Determine the surface area if the edge of the base in (a) is doubled. 1.4 Determine the volume in (b) if the radius is doubled. QUESTION 2 2 The diagram below represents three identical cylindrical logs stacked together. QUESTION3 The diameter of a log is 20 cm and the length of the logs are 150 cm each. 2.1 Determine the total volume of the three logs. 2.2 Determine the surface area of a single log if the length is multiplied by a factor of 3. 3 The municipality uses gutters as shown below to channel water away from buildings. The gutters are solid and in the shape of semi-cylinders. The inner radius is 23 cm and 5cm less than the outer radius. The gutter is 2 m long and is made of concrete. 3.1 Calculate the volume of the solid gutter 3.2 How much water in litre can the gutter hold. Page 57

58 QUESTION Find the total surface area and volume Determine the area of the following circles (4) Circumference =49 7 r Cic [12] QUESTION The City Municipality has ordered special hexagonal pyramid cones to place in the road to warn drivers of unusual hazards. The cones have to be painted with a red reflective paint. The base of the pyramid is a regular hexagon consisting of six equilateral triangles with side of 30cm as shown below: Calculate the area of the base of the pyramid. (3) The hexagonal pyramid cones are shown in the diagram alongside. The slant height, h, is 65cm. h Calculate the surface area of a cone. (3) Page 58

59 30 cm Saxoflo.maths 5.2 A can of aerosol deodorant has a cylindrical shape with a hemisphere on top as shown in the diagram below. The diameter of the cylinder is 6cm and the height of the cylinder is 17cm. QUESTION 6 Calculate the total volume of the can of deodorant. (5) [11] 6.1 A time-capsule in which mementoes will be saved consists of a cylindrical body with a with a hemisphere on top. The diameter of the cylinder is 30 cm and the height of the cylinder is 150 cm. 150 cm Calculate the total volume of the time-capsule. (5) 6.2 A tent manufacturer makes a tent in the shape of a pyramid with a square base. The square ground sheet of the tent is attached to the tent. If slant height, h, is 350 cm and a side of the base is 200 cm, h 200 cm Calculate the total surface area of the tent. (5) [10] Page 59

60 QUESTION 7 A time-capsule is made of a cone and a cylinder and is filled with goods for remembrance. 20 cm Its dimensions are as given in the sketch. 7.1 Calculate the volume of the capsule. (5) 7.2 Calculate the total surface area of the capsule. (5) 50 cm QUESTION 8 15 cm Calculate the volume and total external area of each of the following Page 60

61 8.3 QUESTION 9 V 4 r 3 3 lbh V S 2lb 2bh 2lh 2 S 4 r 4 balls are packed tightly into a square container, such that all sides of the square container touch the balls. The square container has a length 140 mm. Tennis balls Explain why the radius of a ball is 35 mm. Calculate the total volume of the four balls. Calculate the volume of air in the square container that is not taken up by the balls when all four balls are in the container. (2) (4) (4) [10] Page 61

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER)

BY:Prof. RAHUL MISHRA Class :- X QNo. VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) CIRCLES Subject :- Maths General Instructions Questions M:9999907099,9818932244 1 In the adjoining figures, PQ