# 1616 16.1Applications in Two-dimensional Problems 16.2Basic Terminology in Three-dimensional Figures 16.3Applications in Three-dimensional Problems Chapter.

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1616 16.1Applications in Two-dimensional Problems 16.2Basic Terminology in Three-dimensional Figures 16.3Applications in Three-dimensional Problems Chapter Summary Case Study Trigonometry (3)

P. 2 As shown in the figure, M and N are the mid-points of BQ and DQ respectively. Case Study Although PMA is longer than PA, PM and MA are less steep than PA, and hence they are more comfortable to walk up. It is more comfortable to walk up an inclined road along a zigzag path, let me explain it to you. How can we walk up the hill in a relatively more comfortable way? In order to explain the above question, we can use a prism to illustrate the situation such that AB is the top of the inclined road and PQ is the horizontal ground level.

P. 3 When we observe an object above us, the angle  between our line of sight and the horizontal is called the angle of elevation. 16.1 Applications in Two-dimensional Problems Problems When we observe an object below us, the angle  between the line of sight and the horizontal is called the angle of depression. These two angles are important in solving practical trigonometric problems. A. Angle of Elevation and Angle of Depression

P. 4 Example 16.1T In the figure, TB is a flag. The angles of elevation from a point A to the top T and the base B of the flag are 35  and 20  respectively. If the flag is 3 m long, find the distance between A and T. 16.1 Applications in Two-dimensional Problems Problems  TAB  35   20  (cor. to 3 sig. fig.) Solution: A. Angle of Elevation and Angle of Depression  15   TBA  90   20  (ext.  of  )  110  By sine formula,

P. 5 Example 16.2T Solution: 16.1 Applications in Two-dimensional Problems Problems  The height of the platform  (4.5425  1.7) m (cor. to the nearest m) A. Angle of Elevation and Angle of Depression Fanny looks down from a platform at one end of a swimming pool. There is a boy A at the far end of the pool and another boy B in the pool between A and the platform. The boys are 25 m apart and in the same lane. The angles of depression of boy A and boy B from Fanny are 6  and 14  respectively. (a)Find the height of the platform if Fanny’s eyes are 1.7 m above the platform. (a)By sine formula,  ACB  8   CAB  6  (alt.  s, // lines)  CBD  14  (alt.  s, // lines) In  CBD,

P. 6 Example 16.2T 16.1 Applications in Two-dimensional Problems Problems  BD  18.2190 m  18 m (cor. to the nearest m)  Boy B is 18 m from the near end of the pool. A. Angle of Elevation and Angle of Depression Fanny looks down from a platform at one end of a swimming pool. There is a boy A at the far end of the pool and another boy B in the pool between A and the platform. The boys are 25 m apart and in the same lane. The angles of depression of boy A and boy B from Fanny are 6  and 14  respectively. (a)Find the height of the platform if Fanny’s eyes are 1.7 m above the platform. (b)How far is boy B from the near end of the pool? (Give the answers correct to the nearest m.) Solution: (b)In  CBD,  CBD  14  (alt.  s, // lines), BC  18.7767 m

P. 7 16.1 Applications in Two-dimensional Problems Problems B. Bearing In junior forms, we learnt how to use a compass bearing or a true bearing to indicate the direction of an object from a given point. Compass bearing is also known as reduced bearing, and true bearing is also known as whole circle bearing. When using a compass bearing, directions are measured from the north (N) or the south (S), thus the bearing is represented in the form: N  E, N  W, S  E or S  W, where 0     90 . Notes: If   0  or 90 , we simply write it as N, E, S or W. When using a true bearing, all directions are measured from the north in a clockwise direction. The bearing is expressed in the form , where 0     360  and written in three digits such as 007 , 056  or 198 .

P. 8 Example 16.3T 16.1 Applications in Two-dimensional Problems Problems (cor. to 1 d. p.) B. Bearing Eric and Frank are cycling away from P. Eric is cycling in the direction 150  with a speed of 12 m/s and Frank is cycling in the direction 220  with a speed of 10 m/s. After five minutes, they stop and take a rest. (a)What is the distance between them now? (Give the answers correct to 1 decimal place.) Solution: (a)PE  (12  60  5) m  3600 m PF  (10  60  5) m  3000 m  FPE  220   150   70  By cosine formula, 3600 m 3000 m 3817.3767 m

P. 9 Example 16.3T 16.1 Applications in Two-dimensional Problems Problems (cor. to 1 d. p.) True bearing  360   47.6026   30  B. Bearing Eric and Frank are cycling away from P. Eric is cycling in the direction 150  with a speed of 12 m/s and Frank is cycling in the direction 220  with a speed of 10 m/s. After five minutes, they stop and take a rest. (a)What is the distance between them now? (b)Find the true bearing of Frank from Eric. (Give the answers correct to 1 decimal place.) Solution: (b)a  180   150   30 ,b  a  30  (alt.  s, // lines) By sine formula, 3600 m 3000 m 3817.3767 m

P. 10 Example 16.3T 16.1 Applications in Two-dimensional Problems Problems B. Bearing Eric and Frank are cycling away from P. Eric is cycling in the direction 150  with a speed of 12 m/s and Frank is cycling in the direction 220  with a speed of 10 m/s. After five minutes, they stop and take a rest. (a)What is the distance between them now? (b)Find the true bearing of Frank from Eric. (c)After having a rest, if they cycle towards each other at the same speed as before, how long does it take for them to meet? (Give the answers correct to 1 decimal place.) 3600 m 3000 m 3817.3767 m Solution: (c)Time taken (cor. to 1 d. p.)

P. 11 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures A. Terms and Definitions 1. Angle between Two Straight Lines The figure shows two intersecting straight lines lying on the same plane. The acute angle  is called the angle between the two straight lines AB and CD. In 3-D Figures, we can also identify the angle between two straight lines. For example, the angle between BH and FH is  BHF.

P. 12 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures Remark: If the line is perpendicular to the plane, then the projection of the line on the plane is only a point. A. Terms and Definitions 2. Angle between a Straight Line and a Plane When a line is inclined on a plane, we can get a projection of the line on the plane. For example, when a javelin TP hits the ground, the line AP is the projection of TP on the ground. In three-dimensional space, the angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane. For example, the projection of the line AG on AEHD is AH. the angles between the line AG and AEHD is  GAH.

P. 13 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures A. Terms and Definitions 3. Angle between Two Planes Consider the following two situations. (a)A wooden door is opened. (b)A greeting card is standing on a table. In the above two cases, we observe that there are two planes intersecting with each other.

P. 14 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures A. Terms and Definitions When two planes intersect, they meet at a straight line which is called the line of intersection. In the figure,  and  are two planes while AQB is the line of intersection. PS and RT are lines on the planes  and  respectively such that PQ  AB and RQ  AB. The angle  between the lines PQ and RQ is called the angle between planes  and . Remarks: 1.Actually, the angle between two intersecting planes can be acute or obtuse. 2.Usually, we do not consider the reflex angle as the angle between two intersecting planes.

P. 15 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures A. Terms and Definitions The figure shows a rectangular block.  For planes ABFE and BCHE:  Line of intersection:_____________  Angle between 2 planes:_____________ BC  ABE /  DCH  For planes CDEF and EFGH:  Line of intersection:_____________  Angle between 2 planes:_____________ EF  CFG /  D  The figure shows a right pyramid with a square base.  For planes VCD and ABCD:  Line of intersection:_____________  Angle between 2 planes:_____________ CD  VPQ  For planes VBC and VCD:  Line of intersection:_____________  Angle between 2 planes:_____________ VC  BND

P. 16 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures A. Terms and Definitions 4. Distance between a Point and a Straight Line Consider a rectangular pyramid. The distance between the point B and the line VC is the perpendicular distance between B and VC, that is, the length of BE. 5. Distance between a Point and a Plane The distance between a point and a plane is the distance between the point and its projection on the plane, that is, the perpendicular distance between the point and the plane. As shown in the figure, PQ is the distance between point P and the plane.

P. 17 Example 16.4T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (a)In  EFG, (Pyth. theorem) In  AEG, (Pyth. theorem) (b)  FAG is the angle between the lines AG and AF. In  AFG,  FAG  50.2  (cor. to 3 sig. fig.)  The angle between the lines AG and AF is 50.2 . A. Terms and Definitions The figure shows a cuboid. AB  3 cm, AD  6 cm and BF  4 cm. (a)Find the length of AG and express the answer in surd form. (b)Find the angle between the lines AG and AF. (Give the answer correct to 3 significant figures.) Solution:

P. 18 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (a) In  VPD, (Pyth. theorem) In  BCP, (Pyth. theorem) A. Terms and Definitions The figure shows a regular rectangular pyramid with base 12 cm  10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a)Find VP and BP and give the answers in surd form if necessary. Solution: Example 16.5T

P. 19 Example 16.5T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (b)(i)  BVD is the angle between lines VB and VD. In  BCD, (Pyth. theorem) In  BVD, by cosine formula,  BVD  62.8  (cor. to 3 sig. fig.)  The angle between lines VB and VD is 62.8 . A. Terms and Definitions The figure shows a regular rectangular pyramid with base 12 cm  10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a)Find VP and BP and give the answers in surd form if necessary. (b) Find the angle between (i) lines VB and VD,(ii) lines VP and BP. (Give the answers correct to 3 significant figures.) Solution:

P. 20 Example 16.5T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures  BPV  66.9  (cor. to 3 sig. fig.) A. Terms and Definitions The figure shows a regular rectangular pyramid with base 12 cm  10 cm and slant height 15 cm. Suppose P is the mid-point of CD. (a)Find VP and BP and give the answers in surd form if necessary. (b) Find the angle between (i) lines VB and VD,(ii) lines VP and BP. (Give the answers correct to 3 significant figures.) Solution: (b)(ii)  BPV is the angle between lines VP and BP. In  BVP, by cosine formula,  The angle between lines VP and BP is 66.9 .

P. 21 Example 16.6T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (Pyth. theorem) (cor. to 3 sig. fig.) In  CDF, CD  AB  10 cm (cor. to 3 sig. fig.) (b)Since BF is the projection of BD on plane EBCF,  DBF is the required angle.  DBF  17.7  (cor. to 3 sig. fig.) A. Terms and Definitions The figure shows a wedge with rectangular planes ABCD, EFDA and EBCF. AB  10 cm, BC  16 cm,  DCF  35  and DF  CF. (a)Find the lengths of BD and DF. (b)Find the angle between line BD and plane EBCF. (Give the answers correct to 3 significant figures.) (a)In  ABD,  The angle between line BD and plane EBCF is 17.7 . Solution:

P. 22 Example 16.7T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (a)Since EC is the projection of AC on the plane BCFE,  ACE is the required angle. In  CEF, (Pyth. theorem)  17 cm In  ACE,  ACE  19.4  (cor. to 3 sig. fig.)  The angle between the line AC and the plane BCFE is 19.4 . A. Terms and Definitions The figure shows a right-angled triangular prism with ABCD, AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF  6 cm, FC  8 cm and AD  15 cm. Find the angle between (a) the line AC and the plane BCFE; (b) the line AP and the plane BCFE. (Give the answers correct to 3 significant figures.) Solution:

P. 23 Example 16.7T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures In  BEP, BP  7.5 cm. (Pyth. theorem) In  AEP, A. Terms and Definitions The figure shows a right-angled triangular prism with ABCD, AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF  6 cm, FC  8 cm and AD  15 cm. Find the angle between (a) the line AC and the plane BCFE; (b) the line AP and the plane BCFE. (Give the answers correct to 3 significant figures.) Solution: (b)Since EP is the projection of AP on the plane BCFE,  APE is the required angle.  APE  28.7  (cor. to 3 sig. fig.)  The angle between the line AP and the plane BCFE is 28.7 .

P. 24 Example 16.8T Solution: 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (a)  ABC   ACB (base  s, isos.  ) (  sum of  )  45  In  ABM, (b)  VMA is the angle between the planes ABC and VBC. In  AVM,   VMA  60.5  (cor. to 3 sig. fig.) A. Terms and Definitions The figure shows a pyramid with a right-angled triangular base. AB  AC  4 cm, VA  5 cm and  VAB   VAC  90 . (a)Find the length of AM where M is the mid-point of BC in surd form. (b)Find the angle between the planes ABC and VBC. (Give the answer correct to 3 significant figures.)  The angle between the planes ABC and VBC is 60.5 .

P. 25 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures B. Lines of Greatest Slope Notes: There are infinitely many lines of greatest slope on a given inclined plane, such as l 1, l 2, and l 3 (that are parallel to the line PQ) in the figure. In the figure, the inclined plane ABCD intersects the horizontal plane ABEF at the line AB. Three lines XY, PQ and ST are drawn on the inclined plane with PQ perpendicular to AB. Let ,  and  be the angles that XY, PQ and ST make with the horizontal plane respectively. If we compare the three angles, we find that  >  and  > . In fact, PQ makes the largest angle with the horizontal plane and it is called the line of greatest slope of the inclined plane.

P. 26 Example 16.9T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (Pyth. theorem)  0.2828 B. Lines of Greatest Slope The figure shows a right-angled triangular prism with ABCD, BCEF and AFED as rectangles. M and N are the mid-points of BC and AD respectively. If EC  5 cm, DE  2 cm and DA  5DE, find (a) the angle between the BN and plane BCEF, (b) the angle between line NC and plane BCEF, (c) the inclination of the line of greatest slope of plane ABCD. (Give the answers correct to 3 significant figures.) Solution: BF  5 cm and AD  FE  10 cm (a)  NBP is the required angle. P Let P be the mid-point of EF. FP  5 cm and NP  2 cm In  BFP,  The required angle is 15.8 .

P. 27 Example 16.9T 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (cor. to 3 sig. fig.) B. Lines of Greatest Slope The figure shows a right-angled triangular prism with ABCD, BCEF and AFED as rectangles. M and N are the mid-points of BC and AD respectively. If EC  5 cm, DE  2 cm and DA  5DE, find (a) the angle between the BN and plane BCEF, (b) the angle between line NC and plane BCEF, (c) the inclination of the line of greatest slope of plane ABCD. (Give the answers correct to 3 significant figures.) Solution: P (b)  NCP is the required angle. Since  CPN   BPN (SAS).  NCP   NBP (c)Line of greatest slope: CD   DCE is the required angle. (cor. to 3 sig. fig.) 

P. 28 16.3 Applications in Three-dimensional Problems Problems In this section, we shall further study some applications of trigonometric formulas in three-dimensional figures, together with bearings and angles of elevation and depression.

P. 29 Example 16.10T Solution: 16.3 Applications in Three-dimensional Problems Problems In  CDF,In  ACD, (Pyth. theorem)  96.891 m In  ACE,   ACE  5.92  (cor. to 3 sig. fig.)  The inclination of his path with the ground EBCF is 5.92 . The figure shows the plane of a hillside. E is due west of F and C is due south of F. The inclination of the path CD is 8 . DF  10 m and AD  65 m. Eric runs directly from C to A. (a)Find the inclination of his path with the ground EBCF. (b)Find the compass bearing of his path from C. (Give the answers correct to 3 significant figures.) (a)Since EC is the projection of AC on the ground EBCF,  ACE is the required angle.

P. 30 Example 16.10T 16.3 Applications in Three-dimensional Problems Problems In  CDF,  71.1537 m  ECF  42.4122   42.4  (cor. to 3 sig. fig.) Solution: The figure shows the plane of a hillside. E is due west of F and C is due south of F. The inclination of the path CD is 8 . DF  10 m and AD  65 m. Eric runs directly from C to A. (a)Find the inclination of his path with the ground EBCF. (b)Find the compass bearing of his path from C. (Give the answers correct to 3 significant figures.) (b)The projection of AC on the ground EBCF is EC.  The compass bearing of Eric’s path from C is N42.4  W. In  CEF, EF  AD  65 m.

P. 31 Example 16.11T 16.3 Applications in Three-dimensional Problems Problems  335.8846 m (cor. to 3 sig. fig.) A lighthouse VA with height 90 m stands on the same plane as two ships P and Q. The bearings of the lighthouse from P and Q are N50  E and N65  W respectively. The angle of elevation of V from P is 15  and the distance between P and Q is 800 m. (a)Find the distance between P and A. (b)Find the distance between Q and A. (Give the answers correct to 3 significant figures.) Solution: (a)In  VAP,(b)  PAQ  50   65   115  (alt.  s, // lines) By cosine formula, (cor. to 3 sig. fig.) or  882 m (rejected)

P. 32 Example 16.11T 16.3 Applications in Three-dimensional Problems Problems  VQA  8.5607   8.56  (cor. to 3 sig. fig.) A lighthouse VA with height 90 m stands on the same plane as two ships P and Q. The bearings of the lighthouse from P and Q are N50  E and N65  W respectively. The angle of elevation of V from P is 15  and the distance between P and Q is 800 m. (a)Find the distance between P and A. (b)Find the distance between Q and A. (c)Hence find the angle of elevation of V from Q. (Give the answers correct to 3 significant figures.) Solution: (c)In  VAQ,  The angle of elevation of V from Q is 8.56 .

P. 33 16.1 Applications in Two-dimensional Problems Chapter Summary 1.Angles of elevation and depression The angle between the line of sight of an object above us and the horizontal is the angle of elevation. The angle between the line of sight of an object below us and the horizontal is the angle of depression. 2.(a) Compass bearing All directions are measured from the north (N) or the south (S). The bearing is expressed in the form N  E, N  W, S  E or S  W, where 0     90 . (b) True bearing All directions are measured from the north in a clockwise direction. The bearing is expressed in the form , where 0   360  and written in three digits.

P. 34 Chapter Summary 16.2 Basic Terminology in Three-dimensional Figures 1.Angle between Two Straight Lines The angle between two intersecting straight lines is the acute angle formed by the two straight lines lying on the same plane. 2.Angle between a Straight Line and a Plane The angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane. 3.Angle between Two Planes The angle between two planes is the angle between two perpendiculars on the respective planes to the line of intersection of the two planes.

P. 35 Chapter Summary 16.2 Basic Terminology in Three-dimensional Figures 4.Distance between a Point and a Straight Line The distance between a point and a straight line is the perpendicular distance from the point to the line. 5.Distance between a Point and a Plane The distance between a point and a plane is the distance between the point and its projection on the plane. 6.Lines of Greatest Slope If PQ  AB, then PQ is called the line of greatest slope of the inclined plane ABCD.

P. 36 Chapter Summary 16.3 Applications in Three-dimensional Problems In three-dimensional figures, we can also find (a) angles of elevation and depression, and (b) bearing.

Follow-up 16.1 16.1 Applications in Two-dimensional Problems Problems (cor. to 1 d. p.) A. Angle of Elevation and Angle of Depression In the figure, BC is a lighthouse. The angle of depression from B to a point A is 34 , and the angle of elevation from C to A is 40 . If AB  16 m and AC  19 m, find the height of the lighthouse BC. (Give the answer correct to 1 decimal place.) Height of the light house BC  BD  DC Solution: D  (16 sin 34   19 sin 40  ) m  21.2 m Alternative Solution:  BAC  34   40   74  By cosine formula, (cor. to 1 d. p.)  21.2 m

Follow-up 16.2 16.1 Applications in Two-dimensional Problems Problems  41.8  The distance between Jane’s eyes and the ground is 41.8 m. A. Angle of Elevation and Angle of Depression Jane looks down from the top of a building. The angles of depression of a car and a lorry from Jane are 16  and 26  respectively and the two vehicles are 60 m apart. (a)What is the distance between her eyes and the ground? (b)Find the distance between the car and the building. (Give the answers correct to 1 decimal place.) Solution: (a)Let h m be the distance between Jane’s eyes and the ground, r m be the distance between Jane’s eyes and the lorry. By sine formula, h mr m 16  26  (cor. to 3 sig. fig.)

Follow-up 16.2 16.1 Applications in Two-dimensional Problems Problems (cor. to 1 d. p.) A. Angle of Elevation and Angle of Depression Jane looks down from the top of a building. The angles of depression of a car and a lorry from Jane are 16  and 26  respectively and the two vehicles are 60 m apart. (a)What is the distance between her eyes and the ground? (b)Find the distance between the car and the building. (Give the answers correct to 1 decimal place.) Solution: (b)Let a m be the distance between the lorry and the building. h mr m 16  26   Distance between the car and the building  (60  85.6011) m

Follow-up 16.3 16.1 Applications in Two-dimensional Problems Problems  1161.4 m (cor. to 1 d. p.)  The distance between ship A and ship B is 1161.4 m. B. Bearing In the figure, two ships are sailing away from a port P. The true bearing of ship A from the port P is 020  and the ship is 500 m away from the port. The bearing of ship B from the port is 145  and the ship is 800 m away from the port. (a)Find the distance between ships A and B. (b)What is the true bearing of ship B from ship A? (Give the answers correct to 1 decimal place.) Solution: (a)  APB  145   20   125  By cosine formula,

Follow-up 16.3 16.1 Applications in Two-dimensional Problems Problems   PAB  34.4  (cor. to 1 d. p.) Since ship A is 020  from the port,  180   34.4  + 20  (cor. to 1 d. p.) B. Bearing In the figure, two ships are sailing away from a port P. The true bearing of ship A from the port P is 020  and the ship is 500 m away from the port. The bearing of ship B from the port is 145  and the ship is 800 m away from the port. (a)Find the distance between ships A and B. (b)What is the true bearing of ship B from ship A? (Give the answers correct to 1 decimal place.) Solution: (b)By sine formula, the true bearing of ship B from ship A

Follow-up 16.4 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (a) In  FGH, (Pyth. theorem)  17 cm In  DFH, (Pyth. theorem) A. Terms and Definitions The figure shows a rectangular block with dimensions 6 cm  8 cm  15 cm. (a) Find the length of DF and express the answer in surd form. (b) Hence find the angle between (i)lines DF and FH;(ii) lines DF and EF. (Give the answers correct to 3 significant figures.) Solution:

Follow-up 16.4 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures In  DFH,  DFH  19.4  (cor. to 3 sig. fig.)  DFE  63.7  (cor. to 3 sig. fig.) A. Terms and Definitions The figure shows a rectangular block with dimensions 6 cm  8 cm  15 cm. (a) Find the length of DF and express the answer in surd form. (b) Hence find the angle between (i)lines DF and FH;(ii) lines DF and EF. (Give the answers correct to 3 significant figures.) Solution: (b)(i)  DFH is the angle between lines DF and FH.  The angle between lines DF and FH is 19.4 . (ii)  DFE is the angle between lines DF and EF. In  DEF,  The angle between lines DF and EF is 63.7 .

Follow-up 16.5 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (Pyth. theorem) A. Terms and Definitions The figure shows a regular tetrahedron VABC with side 4 cm. M and N are the mid-points of AB and BC respectively. (a)Find the distance between V and line AB. Express the answer in surd form. (b)Find the angle between the lines VM and VN. (Give the answer correct to 3 significant figures.) Solution: (a)VM is the distance between the point V and line AB. In  AVM,  The distance between V and line AB is 2 cm.

Follow-up 16.5 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures In  MNB, by cosine formula,  2 cm  MVN is the angle between the lines VM and VN. (cor. to 3 sig. fig.) A. Terms and Definitions The figure shows a regular tetrahedron VABC with side 4 cm. M and N are the mid-points of AB and BC respectively. (a)Find the distance between V and line AB. Express the answer in surd form. (b)Find the angle between the lines VM and VN. (Give the answer correct to 3 significant figures.) Solution:  The angle between the lines VM and VN is 33.6 .

Follow-up 16.6 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures   BGF  33.7  (cor. to 3 sig. fig.) (Pyth. theorem) Since EG is the projection of AG on plane EFGH,  AGE is the required angle.  AGE  30.2  A. Terms and Definitions The figure shows a rectangular block with dimensions 9 cm  5 cm  6 cm. (a)Find the angle between line BG and plane EFGH. (b)Find the angle between line AG and plane EFGH. (Give the answers correct to 3 significant figures.) Solution: (a)Since FG is the projection of BG on plane EFGH,  BGF is the required angle.  The angle between line BG and plane EFGH is 33.7 . (b)In  EFG,In  AEG,  The angle between line AG and plane EFGH is 30.2 . (cor. to 3 sig. fig.)

Follow-up 16.7 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures (a)In  ARC, (Pyth. theorem) In  APR, A. Terms and Definitions The figure shows a triangular prism. BCFE is a rectangle with dimensions 8 cm  12 cm. DABC is an equilateral triangle with side 8 cm. P and Q are the mid-points of BC and EF respectively. If R is the mid-point of CF, find (a)the length of AR; (b)the angle between the line AR and the plane BCFE. (Give the answers correct to 3 significant figures if necessary.) Solution: (b)Since PR is the projection of AR on the plane BCFE,  ARP is the required angle. In  PCR,  ARP  43.9  (cor. to 3 sig. fig.)  The required angle is 43.9 . R

Follow-up 16.8 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures  20 cm A. Terms and Definitions The figure shows a right pyramid with a square base. BD  30 cm and VD  25 cm. (a)Find the distance between vertex V and plane ABCD. (b)Find the angle between plane VCD and base ABCD. (Give the answers correct to 3 significant figures.) Solution: (a)Let P be the projection of V on the plane ABCD. Then P lies on BD and BP  PD. In  VPD, PD  15 cm. (Pyth. theorem)  The distance between V and the plane ABCD is 20 cm. P

Follow-up 16.8 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures In  VMD, A. Terms and Definitions The figure shows a right pyramid with a square base. BD  30 cm and VD  25 cm. (a)Find the distance between vertex V and plane ABCD. (b)Find the angle between plane VCD and base ABCD. (Give the answers correct to 3 significant figures.) Solution: (b)In  BCD, (Pyth. theorem) PM Let M be the mid-point of DC. In  VPM,  VMP  62.1  (cor. to 3 sig. fig.)  The the angle between plane VCD and base ABCD is 62.1 .

Follow-up 16.9 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures B. Lines of Greatest Slope A zigzag uphill road is built on a right-angled triangular prism as shown in the figure. AB  40 m, DE  20 m and AD  60 m. The road first goes from B to G, and then turns from G to A, where G is the mid-point of CD. Find the angle between (a)the line CD and the plane BCEF, (b)the line BG and the plane BCEF, (c)the line AG and the horizontal ground. (Give the answers correct to 3 significant figures if necessary.) Solution: (a)  DCE is the required angle.

Follow-up 16.9 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures B. Lines of Greatest Slope A zigzag uphill road is built on a right-angled triangular prism as shown in the figure. AB  40 m, DE  20 m and AD  60 m. The road first goes from B to G, and then turns from G to A, where G is the mid-point of CD. Find the angle between (a)the line CD and the plane BCEF, (b)the line BG and the plane BCEF, (c)the line AG and the horizontal ground. (Give the answers correct to 3 significant figures if necessary.) Solution: (b)Let P be the mid-point of CE. P  GBP is the required angle. (cor. to 3 sig. fig.) In  GBP,

Follow-up 16.9 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures B. Lines of Greatest Slope A zigzag uphill road is built on a right-angled triangular prism as shown in the figure. AB  40 m, DE  20 m and AD  60 m. The road first goes from B to G, and then turns from G to A, where G is the mid-point of CD. Find the angle between (a)the line CD and the plane BCEF, (b)the line BG and the plane BCEF, (c)the line AG and the horizontal ground. (Give the answers correct to 3 significant figures if necessary.) P Solution: (c)Let Q be the mid-point of AF. Q  AGQ is the required angle. AQ  FQ  PG  10 m (cor. to 3 sig. fig.) In  AGQ,

Follow-up 16.9 16.2 Basic Terminology in Three- dimensional Figures dimensional Figures B. Lines of Greatest Slope A zigzag uphill road is built on a right-angled triangular prism as shown in the figure. AB  40 m, DE  20 m and AD  60 m. The road first goes from B to G, and then turns from G to A, where G is the mid-point of CD. Find the angle between (a)the line CD and the plane BCEF, (b)the line BG and the plane BCEF, (c)the line AG and the horizontal ground. (Give the answers correct to 3 significant figures if necessary.) (d)Compare these results with the inclination of the line of greatest slope of the plane ABCD. P Solution: (d)From (a),  DCE  30 . From (b) and (c),  GBP  9.10  and  AGQ  9.10 . Q The results in (b) and (c) are less than the inclination of the line of greatest slope, that is,  DCE in (a).

Follow-up 16.10 16.3 Applications in Three-dimensional Problems Problems In  ABP, (Pyth. theorem)  56.8 m (cor. to 3 sig. fig.)  The distance that the girl runs is 56.8 m. The plane of a hillside can be modeled by a triangular prism with EBCF as the ground. F is due east of E and B is due south of E. The gradient of the slope from C to D is 1 : 8. AD  80 m and CF  40 m. Suppose P is the mid-point of AD and a girl runs from B to P. (a)Find the distance that the girl runs. (b)Find the compass bearing of her path in (a). (Give the answers correct to 3 significant figures if necessary.) Solution:

Follow-up 16.10 16.3 Applications in Three-dimensional Problems Problems (b) Let Q be the mid-point of EF. The projection of BP on the ground EBCF is BQ. In  EBQ,   EBQ  45  The plane of a hillside can be modeled by a triangular prism with EBCF as the ground. F is due east of E and B is due south of E. The gradient of the slope from C to D is 1 : 8. AD  80 m and CF  40 m. Suppose P is the mid-point of AD and a girl runs from B to P. (a)Find the distance that the girl runs. (b)Find the compass bearing of her path in (a). (Give the answers correct to 3 significant figures if necessary.) Solution:  The compass bearing of her path is N45  E.

Follow-up 16.11 16.3 Applications in Three-dimensional Problems Problems (a)  BAC  180   40   35  (  sum of  )  105  By sine formula, (cor. to 3 sig. fig.) A vertical pole TA is fixed by two strings on a horizontal plane at B and C. Suppose  ABC  35 ,  ACB  40 , BC  20 m and TC  16 m. (a)Find the lengths of AC and AB. (b)Find the height of the pole. (c)Find the angle of elevation of T from B. (Give the answers correct to 3 significant figures.) Solution:

Follow-up 16.11 16.3 Applications in Three-dimensional Problems Problems (Pyth. theorem)  10.7 m (cor. to 3 sig. fig.)  The height of the pole is 10.7 m. (c) In  TAB,  TBA  38.9  (cor. to 3 sig. fig.)  The angle of elevation of T from B is 38.9 . A vertical pole TA is fixed by two strings on a horizontal plane at B and C. Suppose  ABC  35 ,  ACB  40 , BC  20 m and TC  16 m. (a)Find the lengths of AC and AB. (b)Find the height of the pole. (c)Find the angle of elevation of T from B. (Give the answers correct to 3 significant figures.) Solution: (b) In  TAC,

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