1. HKDSE MATHEMATICS Ronald Hui Tak Sun Secondary School

2. HOMEWORK  SHW6-01, 6-A1  Deadline: 11 Jan 2016 (Monday)  SHW6-B1, 6-C1  Deadline: 18 Jan 2016 (Monday)  No more delay please 22 October 2015 Ronald HUI

3. Book 5A Chapter 6 More Examples on 3-dimensional Problems

4. We have learnt how to solve 3-dimensional problems of the following prisms.

5. Now, let’s try to solve 3-dimensional problems of other kinds of 3-dimensional figures. 40 ∘ A B C D E F 7 cm 6 cm Angle between the planes ACFD and BCFE = ? Angle between the planes VBC and ABCD = ? V A B C D O 10 cm 26 cm 16 cm

6. The figure shows a right triangular prism ABCFDE. Find the angle between the planes ACFD and BCFE, correct to 3 significant figures. The angle between the planes ACFD and BCFE is  ACB. 40 ∘ A B C D E F 7 cm 6 cm Sine formula and cosine formula are useful in solving 3-dimensional problems. In this case, we can find  ACB by the sine formula. 6 cm 40 ∘ A 7 cm

7. Consider △ ABC. sin  ACB BC sin  BAC  AB sin  ACB 7 cm sin 40   6 cm sin  ACB 7 6 sin 40   The figure shows a right triangular prism ABCFDE. Find the angle between the planes ACFD and BCFE, correct to 3 significant figures. (cor. to 3 sig. fig.) 4.33    ACB or 147   (rejected) ∴ The angle between the planes ACFD and BCFE is 33.4 . ∵ 147   40   187   180  40 ∘ A B C D E F 7 cm 6 cm By the sine formula,

8. The figure shows a right triangular prism ABCDEF. Find the angle between the planes ABFE and BCDF, correct to 3 significant figures. Follow-up question A B C D E 10 cm 7 cm 8 cm F The angle between the planes ABFE and BCDF is ∠ ABC. By the cosine formula, ∴ The angle between the planes ABFE and BCDF is 83.3 . AB = EF = 8 cm, BC = FD = 7 cm Consider △ ABC.

9. The figure shows a right pyramid VABCD with a rectangular base. O is the point of intersection of the diagonals AC and BD. It is given that AB = 16 cm, BC = 20 cm and VC = 26 cm. Find the angle between the planes VBC and ABCD, correct to 3 significant figures. First, identify the angle between the planes VBC and ABCD. V A B C D O 20 cm 26 cm 16 cm

10. The figure shows a right pyramid VABCD with a rectangular base. O is the point of intersection of the diagonals AC and BD. It is given that AB = 16 cm, BC = 20 cm and VC = 26 cm. Find the angle between the planes VBC and ABCD, correct to 3 significant figures. Let M be the mid-point of BC. BC is the line of intersection of the planes VBC and ABCD. M V A B C D O 20 cm 26 cm 16 cm ∵ △ VBC and △ BOC are isosceles triangles. ∴ The angle between the planes VBC and ABCD is ∠ VMO. ∴ VM ⊥ BC and OM ⊥ BC

11. The figure shows a right pyramid VABCD with a rectangular base. O is the point of intersection of the diagonals AC and BD. It is given that AB = 16 cm, BC = 20 cm and VC = 26 cm. Find the angle between the planes VBC and ABCD, correct to 3 significant figures. M V A B C D O 20 cm 26 cm 16 cm Then, consider △ VMO. If we know any two sides of this right-angled triangle, we can find ∠ VMO.

12. M V A B C D O 20 cm 26 cm 16 cm OM = 2 16 cm = 8 cm Consider △ VMC. VM VC 2 – MC 2  26 2 – 10 2  cm  24 cm 24 cm 8 cm  cos ∠ VMO VM OM  ∠ VMO 70.5  (cor. to 3 sig. fig.)  Consider △ VMO. ∴ The angle between the planes VBC and ABCD is 70.5 . MC = 2 20 cm = 10 cm ◄ Pyth. theorem

13. The figure shows a right pyramid VABCD of height 15 cm. Its base is a square of side 10 cm. H is the point of intersection of the diagonals AC and BD. Find the angle between the line VA and the plane ABCD, correct to 3 significant figures. The angle between the line VA and the plane ABCD is V A B C D H 10 cm 15 cm Follow-up question ∠ VAH. Step 2 AH is half of AC. Need to find AC. A C H Step 3 Consider △ ABC. AC can be found. AB 10 cm C Step 1 Consider △ VAH. Need to find AH. V HA 15 cm required angle Start from step 3.

14. Consider △ ABC. V A B C D H 10 cm 15 cm AC AB 2 + BC 2  10 2 + 10 2  cm 200  cm ◄ Pyth. theorem ◄ AB = BC = 10 cm ◄ property of square AH  1 2 AC 200  cm 2

15. Consider △ VAH. ∴ The angle between the line VA and the plane ABCD is 64.8 . tan ∠ VAH  VH AH ∠ VAH  64.8  (cor. to 3 sig. fig.) V A B C D H 10 cm 15 cm AH 200  cm 2 15 cm  200 cm 2