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3 Applications of Trigonometry in 3-dimensional Problems


Review Exercise 3 (p. 3.5)1. (a) With the notations in the figure,

(alt. s, QA // BP)

The compass bearing of B from A

(b) With the notations in the figure,

(alt. s, RC // BP)

(

sum of △)

The true bearing of A from C

2. Area of △ABC

3.

∴

∴ Area of △ABC

1

NSS Mathematics in Action 6A Full Solutions

4. Area of △ABC

5.

∴

∴ Area of △ABC

6. ( sum of △)

By the sine formula,

7. By the cosine formula,



10. (a) By the cosine formula,

By the cosine formula,

(b) (property of //gram)

(property of //gram)


∴ The acute angle between the diagonals is 75.

11. Let b 3x m and c 2x m.By the cosine formula,

2


Area of △ABC

ActivityActivity 3.1 (p. 3.10)1. (a) Yes

(b) YP

(c) Slope of , slope of ,

slope of

2. PX. Since FD XY EC and YP is the shortest in length

among the line segments between AB and DC, has the

greatest value.

Quick PracticeQuick Practice 3.1 (p. 3.7)(a) Join BD.

∵ AB and BD intersect at B.∴ The angle between the lines AB and BD is ABD.

(b) Join AF.

∵ AF and EF intersect at F.∴ The angle between the lines AF and EF is AFE.

Quick Practice 3.2 (p. 3.8)(a) (i) ∵ VC is perpendicular to the plane ABC.

∴ C is the projection of V on the plane ABC.∴ AC is the projection of VA on the plane ABC.

(ii) ∵ AC is the projection of VA on the plane ABC.∴ The angle between the line VA and the plane

ABC is VAC.(b) (i) ∵ C is the projection of V on the plane ABC.

∴ MC is the projection of VM on the plane ABC.(ii) ∵ MC is the projection of VM on the plane ABC.

∴ The angle between the line VM and the plane ABC is VMC.

Quick Practice 3.3 (p. 3.9)(a) EF is the line of intersection of the planes CEF and

EFGH.∵ CF EF and GF EF∴ The angle between the planes CEF and EFGH is

CFG.(b) XY is the line of intersection of the planes GHXY and

ABCD. ∵ HX XY and DX XY∴ The angle between the planes GHXY and ABCD is

HXD (or GYC).

3


Quick Practice 3.4 (p. 3.12)(a) ∵ VB VC

∴ △VBC is an isosceles triangle.∵ M is the mid-point of BC.∴ VM BC∴ VM is a line of greatest slope of the inclined plane

VBC.(b) VM is a line of greatest slope of the inclined plane VBC.

∵ VH is perpendicular to the plane ABCD.∴ H is the projection of V on the plane ABCD.∵ HM is the projection of the line VM on the plane

ABCD.∴ The angle between the line of greatest slope of the

inclined plane VBC and the plane ABCD is VMH.(c) BC is the line of intersection of the planes VBC and

ABCD. VM BC∵ M is the mid-point of BC.∴ HM BC∴ The angle between the planes VBC and ABCD is

VMH.

Quick Practice 3.5 (p. 3.13)

(a) (Pyth. theorem)

(b) (i) The angle between BH and AH is AHB.

∴ The angle between BH and AH is 35.3.(ii) The angle between BH and the plane CGHD is

BHC.

∴ The angle between BH and the plane CGHD is 35.3.

Quick Practice 3.6 (p. 3.15)(a) Consider △FBE.

Consider △EBC.

∴

(b) The angle between FB and the plane ABCD is FBD.Consider △FBD.

∴ The angle between FB and the plane ABCD is 21.1.

Quick Practice 3.7 (p. 3.15)The angle between the planes ADFC and BEFC is ACB.Consider △ABC.

∴ The angle between the planes ADFC and BEFC is 28.1.

Quick Practice 3.8 (p. 3.16)(a) The angle between the planes ABGH and ABCD is HAD.

Consider △HAD.

∴ The angle between the planes ABGH and ABCD is 45.

(b)

RS is the line of intersection of the planes ABGH and EFQP.∵ AR RS and PR RS∴ ARP is the angle between the planes ABGH and

EFQP.Consider △EAP.

Consider △ARP.( sum of △)

∴ The angle between the planes ABGH and EFQP is 78.7.

4


Quick Practice 3.9 (p. 3.18)(a) The angle between VB and the plane ABCD is VBO.

Consider △ABD.

∴

Consider △VOB.

∴ The angle between VB and the plane ABCD is 54.06.(b) AB is the line of intersection of the planes VAB and ABCD.

Let M be the mid-point of AB.

∵ △VAB and △OAB are isosceles triangles.∴ VM AB and OM AB∴ The angle between the planes VAB and ABCD is

VMO.

Consider △VOM.

∴ The angle between the planes VAB and ABCD is 62.30.

(c) VO is the line of intersection of the planes VAO and VBO. ∵ AO VO and BO VO∴ The angle between the planes VAO and VBO is

AOB.Consider △AOB.

(property of rectangle)


∴ The angle between the planes VAO and VBO is 87.21.

Quick Practice 3.10 (p. 3.25)(a) Consider △LOA.

Consider △LOB.

5


Consider △OAB.

(b) Consider △OAB.

∴ The true bearing of B from A is (90 + 22.8) = 112.8.

Quick Practice 3.11 (p. 3.27)(a) Consider △BOK.

(Pyth. theorem)

Consider △HOK.

∵ K is due east of H.

∴ KHO 90

∴ The compass bearing of B from O is N53.9E.

(b)

Consider △AOH.

(c) With the notations in the figure,

6


AC (6.076 77 – 5) km 1.076 77 km

Consider △ACB.

The angle of depression of B from A

Quick Practice 3.12 (p. 3.29)(a) Consider △FBD.

Consider △BCD.

(b) The angle of inclination of the plane ABEF is EBC.Consider △EBC.

∴ The angle of inclination of the plane ABEF is 47.6.(c) Let H be the projection of G on the plane ABCD.

DH : HC 2 : 1Consider △BCD.

The true bearing of G from B

7


Quick Practice 3.13 (p. 3.31)(a)

Consider △BFC.By the cosine formula,

(b) (Pyth. theorem)

(Pyth. theorem)

Consider △BCE.By the cosine formula,

(c) Let G be a point on BC such that EG BC and FG BC.The angle between the planes BCE and BCF is EGF.

Consider △BGE.

Consider △EGF.

∴ The angle between the planes BCE and BCF is 75.3.

Further Practice (p. 3.18)1. (a) The angle between HB and BD is HBD.

(Pyth. theorem)

∴ The angle between HB and BD is 25.1.

8


(b) The angle between HB and the plane HDCG is BHC.

(Pyth. theorem)

∴ The angle between HB and the plane HDCG is 34.4.

2. (a)

(b) Consider △AMB.

(c) The angle between the planes DBC and BCFE is DMN.Consider △DMN.

∴ The angle between the planes DBC and BCFE is 23.6.

3. (a)

Consider △ABM.

(b) Let N be the mid-point of AF.

Consider △ANM.

(c) BC is the line of intersection of the planes ABC and BFC.∵ AM BC and FM BC∴ The angle between the planes ABC and BFC is

AMF.

Consider △AFM.

∴ The angle between the planes ABC and BFC is 109.

Further Practice (p. 3.32)1. Let h m be the height of the tower TO.

Consider △TAO.

Consider △TBO.

Consider △OAB.

9


∴ The height of the tower TO is 258 m.

2. (a) Consider △DCQ.

Area of BPQC

(b)

Consider △ABP.

Let F be a point on BC such that BC FP.Consider △BFP.

Area of BPQC

3. (a)

Consider △QBC in Figure (b).By the cosine formula,

(b) Consider △QBC in Figure (b).By the cosine formula,

(c) Let D be a point on BC such that PD BC and QD BC.

10


The angle between the planes BCP and BCQ is PDQ.Consider △PQB.

Consider △QBD.

Consider △PQD.

∴ The angle between the planes BCP and BCQ is 61.4.

Exercise

Exercise 3A (p. 3.19)Level 11. (a) ∵ AH and CH intersect at H.

∴ The angle between AH and CH is AHC.(b) ∵ AG and DG intersect at G.

∴ The angle between AG and DG is AGD.(c) ∵ D is the projection of A on the plane CGHD.

∴ HD is the projection of AH on the plane CGHD.∴ The angle between AH and the plane CGHD is

AHD.(d) ∵ E is the projection of A on the plane EFGH.

∴ EH is the projection of AH on the plane EFGH.∴ The angle between AH and the plane EFGH is

AHE.(e) BC is the line of intersection of the planes BCGF and

BCHE.∵ FB BC and EB BC (or GC BC and

HC BC)∴ The angle between the planes BCGF and BCHE

is EBF (or HCG).(f) AE is the line of intersection of the planes ACGE and

ABFE.∵ EG AE and EF AE (or AC AE and

AB AE)∴ The angle between the planes ACGE and ABFE

is GEF (or CAB).

2. (a) ∵ O is the projection of V on the plane ABCD.∴ OA is the projection of VA on the plane ABCD.∴ The angle between VA and the plane ABCD is

VAO (or VAC).(b) ∵ O is the projection of V on the plane ABCD.

∴ ON is the projection of VN on the plane ABCD.∴ The angle between VN and the plane ABCD is

VNO.(c) BC is the line of intersection of the planes VBC and

ABCD.∵ VN BC and ON BC∴ The angle between the planes VBC and ABCD is

VNO.(d) ∵ VM AB and VN BC

∴ VM and VN are the lines of greatest slope of the inclined planes VAB and VBC respectively.

∵ VM and VN intersect at V.∴ The angle between the lines of greatest slope of

the inclined planes VAB and VBC is MVN.

3. (a)

(b) The angle between BG and the plane ABCD is GBC.

∴ The angle between BG and the plane ABCD is 45°.

(c) The angle between BH and the plane BCGF is HBG.

∴ The angle between BH and the plane BCGF is 35.3°.

4. (a) The angle between AG and GC is AGC.

∴ The angle between AG and GC is 63.4°.

11


(b) The angle between AG and the plane ADHE is GAH.

∴ The angle between AG and the plane ADHE is 45.7°.

5. (a) The angle between HB and DB is HBD.

∴ The angle between HB and DB is 24.9°.(b) The angle between HB and the plane CGHD is

BHC.

∴ The angle between HB and the plane CGHD is 31.8°.

6. The angle between the planes ACFD and BCFE is ACB.(a) In △ABC, by the sine formula,

∴ The angle between the planes ACFD and BCFE is 44.5°.

(b) In △ABC, by the cosine formula,

∴ The angle between the planes ACFD and BCFE is 41.4°.

12


7. (a) The angle between the planes ABQP and ACRP is BAC.Consider △ABC.By the cosine formula,

∴ The angle between the planes ABQP and ACRP is 106°.

(b)

Let M be the mid-point of QR.∵ △AQR and △PQR are isosceles triangles.∴ AM QR and PM QR∴ The angle between the planes AQR and PQR is

AMP.

Consider △PQM.

∴ The angle between the planes AQR and PQR is 73.3°.

8. (a) Consider △CBE.

Consider △DBC.

13


(b) The angle between the path BD and the plane ABEF is DBF.Consider △DBF.

∴ The angle between the path BD and the plane ABEF is 16.7°.

9. (a) Consider △EBF.

Consider △EBC.

(b) The angle between BF and the plane ABCD is FBD.Consider △EBF.

Consider △FBD.

∴ The angle between BF and the plane ABCD is 18.9°.

10. (a) The angle between VA and the plane ABCD is VAN.

∴

Consider △VAN.

∴ The angle between VA and the plane ABCD is

67.4°.

14


(b)

AB is the line of intersection of the planes VAB and ABCD.Let M be the mid-point of AB.∵ △VAB and △NAB are isosceles triangles.∴ VM AB and NM AB∴ The angle between the planes VAB and ABCD is

VMN.

Consider △VMN.

∴ The angle between the planes VAB and ABCD is 76.0°.

11. (a) The angle between VC and the plane ABCD is VCM.

Consider △BCM.

Consider △VCM.

∴ The angle between VC and the plane ABCD is 56.1°.

(b) Let N be the mid-point of CD.

The angle between the planes VDC and ABCD is VNM.Consider △VNM.

∴ The angle between the planes VDC and ABCD is 63.4°.

15


12. (a)

(b) Consider △ABF.By the cosine formula,

(c) The angle between BF and DF is BFD.Consider △BDF.By the cosine formula,

∴ The angle between BF and DF is 77.0°.

13. (a) Consider △BCD.

Consider △VBK.

Height of the cottage

(b)

Let M and N be the mid-points of AD and FE respectively.The angle between the planes VAD and ADEF is VMN.

Consider △VMK.

∴ The angle between the planes VAD and ADEF is 114°.

Level 214. (a) The angle between the planes AFGD and ABCD is

FAB.Consider △FAB.

∴ The angle between the planes AFGD and ABCD is 33.7°.

(b) The angle between the planes ABCD and EPQH is EPA.∵ P is the mid-point of AB.

∴

Consider △EAP.

∴ The angle between the planes ABCD and EPQH is 53.1°.

15. (a) Consider △EFG.

Consider △AEG.

∵ ABGH is a rectangle.

∴ (property of rectangle)

16


Consider △KGH.

(b) ∵ The angle between the planes KGH and EFGH the angle between the planes ABGH and EFGH

∴ The angle between the planes KGH and EFGH is BGF.

Consider △BFG.

∴ The angle between the planes KGH and EFGH is 37.9°.

16. (a) The angle between the covers ABCD and BCEF is DCE.Consider △DCE.

∴ The angle between the covers ABCD and BCEF is 18.4°.

(b)

The angle between BD and the plane BCEF is DBG.Consider △BCD.

Consider △DCG.

Consider △DBG.

∴ The angle between BD and the plane BCEF is 14.3°.

17


17. (a)

Consider △ABM.

(b)

Consider △AMD.By the cosine formula,

(c) The angle between the planes ABC and EBC is AME.

∴ The angle between the planes ABC and EBC is 141°.

18. (a) Consider △ABH.

Consider △ACH.

Consider △ABC.By the cosine formula,

(b) The angle between HM and the plane ABC is HMA.Consider △AMC.By the cosine formula,

18


Consider △HMA.

∴ The angle between HM and the plane ABC is 43.6°.

19. (a) The angle between DC and the plane ABC is DCB.Consider △DCB.

∴ The angle between DC and the plane ABC is 53.1°.

(b) Let M be the mid-point of AC.

∵ △ADC and △ABC are isosceles triangles.∴ DM AC and BM AC∴ The angle between the planes ADC and ABC is

BMD. Consider △DCB.

Consider △BCM.

Consider △DBM.

∴ The angle between the planes ADC and ABC is 62.1°.

20. (a) Let G and H be the mid-points of BC and EF respectively.

The angle between the planes ABC and AEF is GAH.Consider △ABG.

19


Consider △AGH.

∴ The angle between the planes ABC and AEF is 55.8°.

(b)

RS is the line of intersection of the planes AEF and DMN.Let P be the mid-point of RS.∵ DP RS and HP RS∴ The angle between the planes AEF and DMN is

DPH.DHP GAH 55.795° (alt. s, DH // AG)Let O be the mid-point of MN.

Consider △OHD.

∴ The angle between the planes AEF and DMN is 87.9°.

21. (a) The angle between the planes ABDC and CDFE is ACE.Consider △ABE.

Consider △AEC.

20


∴ The angle between the planes ABDC and CDFE is 75.5°.

(b)

Let G be the projection of E on the plane ABDC.∴ CGE 90° and BGE 90°The angle between BE and the plane ABDC is EBG.Consider △CGE.

Consider △BEG.

∴ The angle between BE and the plane ABDC is 50.8°.

22. (a) VN is the line of intersection of the planes VAN and VBN.∵ AN VN and BN VN∴ The angle between the planes VAN and VBN is

ANB.∴ (property of square)∴ The angle between the planes VAN and VBN is

90°.(b) AB is the line of intersection of the planes VAB and

ABCD.Let E be the mid-point of AB.

∵ △VAB and △NAB are isosceles triangles.∴ VE AB and NE AB∴ The angle between the planes VAB and ABCD is

VEN.

Consider △VEN.

21


∴ The angle between the planes VAB and ABCD is 71.6°.

(c)

Let F be the mid-point of DC.The angle between the planes VAB and VCD is EVF.By similar argument as in (b),VFN 71.565° Consider △VEF.

∴ The angle between the planes VAB and VCD is 36.9°.

23. (a)

Let M be the mid-point of AF.

(property of square)

(b) (i) The angle between AB and BF is ABF.Consider △ABF.

22


∵ cos ABF is independent of a.∴ The angle between AB and BF, i.e. ABF,

will not change if the value of a changes.(ii) Let N be the mid-point of BC.

The angle between the planes ABC and FBC is ANF.

Consider △ANF.

∵ cos ANF is independent of a.∴ The angle between the planes ABC and

FBC, i.e. ANF, will not change if the value of a changes.

24. (a)

∵ AP = QC and AF = CH∴ FP HQAlso, FE HE∴ FE FP HE HQ

PE = QEConsider △PQE.

23


Consider △HCQ.

(b) With the notations in the figure,

The angle between the planes ACQP and ABCD is QMN.Consider △ACD.

∵ PACQ is an isosceles trapezium.

∴

Consider △QMC.

Consider △QMN.

∴ The angle between the planes ACQP and ABCD is 65.4°.

Exercise 3B (p. 3.33)Level 11. (a) ∵ DF is perpendicular to the plane BCFE.

∴ F is the projection of D on the plane BCFE.∴ The projection of BD on the plane BCFE is BF.

(b) Consider △CDF.

Consider △BFD.

∴ The distance the car travelled is 40.8 m.

2. (a) Consider △ACB.

Consider △TBA.

∴ The height of the tower TA is 31.5 m.(b) Consider △ACB.

Consider △TCA.

∴ The angle of elevation of T from C is 32.2.

3. (a)

Consider △TAC.

Consider △TAB.

24


Consider △ACB.

(b)

With the notation in the figure, consider △ACB.

∴ The compass bearing of B from C is N64.2W.

4. (a) Consider △ABC.


Consider △XAC.

∴ The angle of elevation of X from A is 14.3.(b) Consider △ABC.


25


Consider △XBC.

∵ The angle of depression of B from X the angle of elevation of X from B

∴ The angle of depression of B from X is 22.2.

5. (a) Consider △ABP.

∴ The height of the tower AB is 69.9 m.

(b)

Consider △BQP.

Consider △ABQ.

∴ The angle of elevation of A from Q is 17.1.

6. (a) Consider △OAB.

Consider △ODC.

Consider △AOD.

26


∴ The compass bearing of C from O is N50.9E.

(b) Distance travelled by the helicopter

Consider △AOD.

7. (a) Consider △TAC.

Consider △TBC.

Consider △ACB.By the cosine formula,

∴ The distance between A and B is 75.9 m.(b)

With the notations in the figure,

27


Consider △ACB.By the cosine formula,

Reflex

∴ The true bearing of A from B is 335.

8. (a) Consider △TAO.

Consider △OAC.

∴ The true bearing of C from A is 39.7.(b) Consider △OAC.

Consider △TCO.

∴ The angle of elevation of T from C is 34.8.(c) Consider △OAB.


Consider △TBO.

∴ The angle of elevation of T from B is 41.8.

28


9. (a)

Consider △AED.By the cosine formula,

∴ The distance between A and D is 1.57 cm.

(b)

Consider △AFD.By the cosine formula,

10. (a)

Consider △AMC.

Consider △ACD.

∴ The length of the stick AD is 96.3 cm.(b) Consider △AMC.

Consider △AMD.

29


Level 211. (a) Consider △EBC.

Consider △BCD.

Consider △EBF.

(b) Consider △FDB.

∴ The angle of inclination of FB is 47.5.(c) ∵ G is the mid-point of EF.

∴

Consider △GBE.

Let H be the projection of G on the plane ABCD.

Consider △BGH.

∴ The angle of inclination of GB is 51.5.

12. (a) Consider △TAG.

Consider △TBG.

(b) Consider △ABG.

(c)

With the notation in the figure,

∴ The compass bearing of B from A is S29.9E.

30


13. (a)

With the notation in the figure,Let AE be the length of the shadow of Tom.

(corr. sides, ~△s)

∴ The length of the shadow of Tom is 4.8 m.(b)


Consider △CDF.

∴ The length of his shadow CD is 2.82 m.(c) The shortest length of Tom’s shadow is attained when

the distance between the lamppost and Tom is the shortest.Let G be a point on AB, which represents where Tom stands.∴Consider △OAB.

Consider △OGB.

Let GH be the length of the shadow of Tom.

(corr. sides, ~△s)

(cor. to 3 sig. fig.)

∴ The shortest length of Tom’s shadow is 2.54 m.

14. (a) Consider △PAO.

31


Consider △PBO.

Consider △PCO.

(b) (i) Consider △OAB.By the cosine formula,

(ii) Consider △OAC.By the cosine formula,

(c) ∵ OAB = OAC∴

32


Consider △OAC.By the cosine formula,

∴ The compass bearing of C from O is N52.5E.

15. (a) Consider △XAC.

∴ The height of the tower XC is 49.7 m.

(b)

Consider △CAB.By the cosine formula,

Consider △XBC.

∴ The angle of elevation of X from B is 41.5.(c)


Consider △CAB.

33


∴ The true bearing of B from C is 152.

16. (a)

Let be the angle of the sector C OC.

∴ The angle of the sector is 120.

(b)

Consider △BC’O.


Consider △BOA.


The length of the ribbon

17. (a)

△PBC is an equilateral triangle.

∴

(b) Let D be a point on BC such that and .

The angle between the plane ABC and the horizontal table is ADP.

Consider △BPD.

34


Consider △ADP.

∴ The angle between the plane ABC and the horizontal table is 63.4.

∵ The value of tan ADP is a constant.∴ The angle will not change if the value of k

changes.

18. (a) Let F and G be the mid-points of AD and BC respectively.

The angle between the planes AED and ABCD is EFG.

Consider △FEG.

∴ The angle between the planes AED and ABCD is 64.1.

(b) (i)

Consider △BEC.By the cosine formula,

Consider △BMC.By the cosine formula,

35


(ii) Consider △ABM.

Revision Exercise 3 (p. 3.39)Level 11. (a) The angle between AF and CF is AFC.

Similarly, ∵ AF = CF = AC∴ △AFC is an equilateral triangle.∴ AFC = 60∴ The angle between AF and CF is 60.

(b) The angle between AF and the plane BCGF is AFB.Consider △AFB.

∴ The angle between AF and the plane BCGF is 45.

(c) The angle between AG and the plane BCGF is AGB.

Consider △AGB.

∴ The angle between AG and the plane BCGF is 35.3.

(d) The angle between the planes AHB and ADHE is 90.

2. (a) Consider △AEF.

Consider △AGF.

36


Consider △GHE.

(b) Let

∴

Area of △AEG

3. (a) The angle between BD and BA is ABD.Consider △ABD.

∴ The angle between BD and BA is 51.3.(b) The angle between BD and the plane BCEF is DBE.

Consider △CDE.

Consider △ABD.

Consider △BDE.

∴ The angle between BD and the plane BCEF is 6.23.

4. (a)

Let Q be the projection of P on the plane ABCD.The angle that PA makes with the base ABCD is PAQ.

37


Consider △ABC.

Consider △PQA.

∴ The angle that PA makes with the base ABCD is 26.6.

(b)

Let M and N be the mid-points of AB and GF respectively.The angle between the planes PAB and ABFG is PMN.

Consider △PMN.

∴ The angle between the planes PAB and ABFG is 58.0.

5. (a) (i) One of the longest line segment is AG.

(ii) The angle between AG and the plane ABCD is GAC.

∴ The required angle is 54.2.

38


(b) (i)

One of the longest line segment is AV.


(ii) The angle between AV and the plane ABCD is VAO.

∴ The required angle is 62.1.

6. ∵ △DEF is an equilateral triangle.∴Let M be a point on EF such that XM EF and DM EF.

Consider △DFM.

Consider △XDM.

7. (a) Consider △ABC.( sum

of △)

39



Consider △PBA.

(b) Area of △ABC

(c) Volume of the tetrahedron PABC

8. (a) Consider △TBP.

Consider △TAP.

Consider △BAP.BPA = 200 150

= 50By the cosine formula,

40


(b)


Consider △BAP.By the cosine formula,

∴ The true bearing of A from B is 107.

9. (a) Consider △ABP.

(b)

Consider △ABD.

With the notations in the figure.

∴ The compass bearing of P from D is S41.3°W.

41


(c) (i) Consider △BCP.

∴ The angle of elevation of P from C is 38.5°.

(ii)

∴ The angle of elevation of P from D is 33.5°.

10. (a)

With the notations in the figure,AG BH 800 mConsider △ACG.

Consider △CHG.

∴ The speed of the aeroplane

(b) Consider △CHG.

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Consider △BCH.

∴ The angle of elevation of B from C is 22.2°.

11. Let O be a point inside the pentagon such that OA = OB = OC = OD = OE.


Consider △AOX.

The height of the pyramid formed

12. The student is not correct. The required angle should be an angle formed by two lines on each of the planes, and the two lines should be perpendicular to the line of intersection of the two planes. Since AB and BF are not perpendicular to the line of intersection, the angle between them is not the angle between the planes ABE and FBE.

Level 213. (a) Consider △DCE.

(b)

Let Q be the projection of P on the plane BCEF.The angle between CP and the plane BCEF is PCQ.

Consider △PCD.

Consider △PCQ.

∴ The angle between CP and the plane BCEF is 12.7°.

14. (a)

Consider △VAB.

(b)

(c) ∵ △VAB △VCB∴ CN VB and CN = ANThe angle between the planes VAB and VBC is ANC.

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Consider △NAC.

∴ The angle between the planes VAB and VBC is 106°.

15. (a) Consider △BCD.

(b) The angle of inclination of FB is DBF.

(i) When ,

∴ The angle of inclination of FB is 23.4 when .

(ii) When ,

∴ The angle of inclination of FB is 25.8 when .

(c) No.∵ The angle of inclination of FB is DBF and

∴ The angle of inclination of FB does not vary inversely as .

16. (a) Consider △ABF.

Consider △BFC.

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(b) Consider △ABF.

Consider △ABC.

Consider △AFC.

17. (a) Let E be a point on CD such that AE CD and BE CD.

The angle between the planes ACD and BCD is AEB.

Consider △ABE.

∴ The angle between the planes ACD and BCD is 53.4°.

(b) Let F be a point on MN such that AF MN and BF MN.AFE is a straight line.

The angle between the planes AMN and BMN is AFB.

Consider △BFE.By the cosine formula,

Consider △AFB.By the cosine formula,

∴ The angle between the planes AMN and BMN is 107°.

18. (a)

Total distance travelled by the man

(b) Let G and H be the projection of X and Y on the plane BCFE respectively.Let Z be a point on YH at the same horizontal level as X.

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The angle between the path XY and the horizontal ground is YXZ.Consider △XCG.

Consider △YBH.

Consider △XYZ.

∴ The angle between the path XY and the horizontal ground is 13.6°.

19. (a) Let M be the mid-point of BC and N be the projection of M on AP.The angle that the sheet makes with the horizontal ground is AMN.Consider △ABM.

Consider △AMN.

∴ The angle that the sheet makes with the horizontal ground is 11.8°.

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(b)

Area of the shadow △PQR

(c) (i) No(ii) The area of the shadow of the sheet is equal to

the area of △PQR, as the height and the base of the triangular shadow do not change, even though the sun shines from the west at a certain angle of elevation.

20. (a) The angle between AC and the horizontal ground is CAE.Consider Figure (a).∵ ADB AEC 90°∴ BD // CE (corr. s equal)∵ BD // CE and AB BC 5 cm∴ AD DE (intercept theorem)

∴

Consider △DAE in Figure (b).By the cosine formula,

Consider △CAE in Figure (b).

∴ The angle between AC and the horizontal ground is 65.5°.

(b) Consider △CAE in Figure (b).

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Consider △ACB in Figure (b).By the cosine formula,

21. (a) With the notation in the figure,

Consider △HPK.

(b) Consider △APH.

Consider △BPK.

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(c)


Consider △ABR.

∵ The angle of depression of B from A the angle of elevation of A from B

∴ The angle of depression of B from A is 25.7°.

22. (a) Consider △DOC.

Consider △DAC.

Consider △COA.

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(b) Consider △DBC.

Consider △COB.

(c) ∵ COA = COB∴ cos COA cos COB

(d)

∴ The compass bearing of B from O is N19.5°E.

23. (a) The angle between DF and BF is DFB.Consider △BFG.

Consider △DFE.

Consider △DAB.

Consider △DFB.By the cosine formula,

∴ The angle between DF and BF is 68.9°.

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(b) Let N be the projection of M on the plane EFGH.

The angle between FM and the plane EFGH is MFN.

∴

Consider △MFN.

∴ The angle between FM and the plane EFGH is 46.7°.

(c) DB is the line of intersection of the planes BDF and BCD.∵ △DFB and △BCD are isosceles triangles.∴ FM DB and CM DB∴ The angle between the planes BDF and BCD is

FMC.Consider △MFN.

∴ The angle between the planes BDF and BCD is 133°.

24. (a) With the notation in the figure,

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Consider △DCF.By the cosine formula,


∴ The required angle of inclination

(b)

Let K be the projection of D on the horizontal table.

∴ The angle of inclination of DE is 20.2°.

25. (a) Consider △AMD.

Let E be the projection of D on the plane ABC.

Consider △DEM.

∴ The height of D above the plane ABC is 5.20 cm.

(b) The angle between AD and the plane ABC is DAE.Consider △DAE.

∴ The angle between AD and the plane ABC is 25.7°.

(c)

Let F be the projection of C on the plane ABD.

The angle between BC and the plane ABD is CBF.∵ MC < MD∴ CF < DE∴ sin CBF < sin DAE∴ CBF < DAE∴ The angle between BC and the plane ABD is less

than that in (b).

26. (a)

∵ △DEQ ~△DFS ~△DBC (AAA)

∴ DE : EF : FB

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∴

(b) (i) (corr. sides, ~△s)

∴ The distance between D and F is 20.5 cm.(ii) Consider △DEF.

(iii) Let M be the projection of B on the plane PRSQ.

The angle between BF and the plane PRSQ is BFM.

Consider △BFM.

∴ The angle between BF and the plane PRSQ is 43.8°.

Multiple Choice Questions (p. 3.45)1. Answer: C

FG is the line of intersection of the planes AFGD and

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EFGH.∵ AF FG and EF FG, DG FG and HG FG∴ The angles between the planes AFGD and EFGH are

AFE or DGH.∴ The answer is C.

2. Answer: A


3. Answer: D

Let M be the intersection of AG and BH.∵

∴

Consider △MAB.

∴ The answer is D.

4. Answer: CLet a be the length of the side of square ABCD.Consider △DCB.

Consider △CBE.

Consider △DBF.

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5. Answer: ALet N be the mid-point of DE.

N is the projection of M on the plane CHED.The angle between MH and the plane CHED is MHN.

Consider △NHE.

Consider △MHN.

∴ The angle between MH and the plane CHED is 32.

6. Answer: ALet J be the mid-point of BC.

The angle between the planes HBC and ABCD is HJK.

Consider △HAK.

Consider △HJK.

∴ The angle between the planes HBC and ABCD is 29.

7. Answer: A

Consider △AMN.

8. Answer: CConsider △ABC.

Consider △ABD.

9. Answer: BWith the notation in the figure,

Consider △ABQ.

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Consider △ABP.

Consider △PQB.

∴ The bearing of A from Q is S30.4E.

10. Answer: B

Area of △PCR

Volume of the tetrahedron CPQR

Investigation Corner (p. 3.49)

(a) (i)

Consider △MXS.

∵∴ The maximum value of is attained when

L = 1.5.Let max be the maximum value of .

∴ The maximum value of is 67.4.

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(b) (i)

∵∴ is larger than .

(ii) ∵∴ The minimum value of L is attained when

= 70.Let Lmin be the minimum value of L.

∴ The minimum value of L is 0.546.

(c) (i) ∵

∴ , where

∴

(ii) XR or XQ is the longest distance of a point on the screen from the projector.

When A = 0.5,

Set

∴ The maximum value of L is 5.36.(d)

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