I.B. Geometry. 3d- Figures min 218 marks

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1 I.. Geometry 3d- Figures min 218 marks 1. In the diagram below, PQRS is the square base of a solid right pyramid with vertex V. The sides of the square are 8 cm, and the height VG is 12 cm. M is the midpoint of [QR]. V VG = 12 cm P G 8 cm M Q S 8 cm R (i) Write down the length of [GM]. (ii) alculate the length of [VM]. Find (i) (ii) the total surface area of the pyramid; the angle between the face VQR and the base of the pyramid. (4) (Total 6 marks) handan s Sample Papers : Sure Success and Excellent Marks 1

2 2. The figure shows two adjacent triangular fields and where = 30 m, = 80 m, = 50m. ˆ = 60 and  = m m 30 m 30 Note: iagram not drawn to scale Using triangle calculate the length. alculate the size of ˆ. (Total 8 marks) 3. The coordinates of the vertices of a triangle are P ( 2, 6), Q (6, 2) and R ( 8, a). On graph paper, mark the points P and Q on a set of coordinate axes. Use 1 cm to represent 1 unit on each axis. (i) alculate the distance PQ. (ii) (iii) Find the gradient of the line PQ. If angle RPQ is a right angle, what is the gradient of the line PR? (1) (iv) Use your answer from (iii), or otherwise, to find the value of 'a'. The length of PR is 180. Find the area of triangle PQR. (Total 13 marks) 4. The diagram below shows an equilateral triangle, with each side 3 cm long. The side [] is extended to so that = 4 cm. 4 cm alculate, correct to two decimal places, the length of [].(Total 4 marks) handan s Sample Papers : Sure Success and Excellent Marks 2

3 5. Raul, in house R, is directly across the lake from Sylvia, in house S. The houses are two kilometres apart. When both Raul and Sylvia are facing due north, they see a speedboat in the lake between the two houses. Raul in house R can see the boat at 35 east of where he is facing. Sylvia in house S can see the same boat at 65 west of where she is facing. opy and complete the diagram below, indicating which is the 35 angle, and which is the 65 angle. N N diagram not to scale R 2 km S (i) alculate the size of R ˆ S. (ii) t this moment, how far is the boat () from Raul's house (R)? Please give your answer to the nearest 100 metres. (5) Raul and Sylvia then see a sailboat on the lake at point Q, which is 2.6 km from Raul (R) and 3.5 km from Sylvia (S). alculate the size of R Qˆ S at that moment, giving your answer to the nearest degree. (4) (Total 11 marks) 6. O is a square based pyramid of side 4 cm as shown in the diagram. The vertex is 3 cm directly above X, the centre of square O. M is the midpoint of. Find the length of XM. alculate the length of M. alculate the angle between the face and the base O. iagram not to scale X M O (Total 8 marks) handan s Sample Papers : Sure Success and Excellent Marks 3

4 7. The diagram below shows the line with equation 3x + 2y = 18. The points and are the y and x-intercepts respectively. M is the midpoint of []. y 3 x+ 2 y=18 M O x Find the coordinates of the point ; the point ; the point M. (Total 8 marks) 8. The diagram below shows a crane PQR that carries a flat box W. (PQ) is vertical, and the floor (PM) is horizontal. R 7.8 m 6.5 m Q 102º W 11.1 m h P M Given that PQ = 11.1m, QR = 7.8m, P Qˆ R =102 and RW = 6.5 m, calculate PR; angle P Rˆ Q; the height, h, of W above (PM). (Total 7 marks) 9. The diagram shows a cuboid 22.5 cm by 40 cm by 30 cm. handan s Sample Papers : Sure Success and Excellent Marks 4

5 H G E F 40 cm alculate the length of []. alculate the size of G Â cm 30 cm (Total 4 marks) 10. The following diagram shows a sloping roof. The surface is a rectangle. The angle E is 55. The vertical height, F, of the roof is 3 m and the length is 7 m. 3 m 7 m 55 E F alculate. alculate the length of the diagonal. (Total 8 marks) 11. The following diagram shows the rectangular prism EFGH. The length is 5 cm, the width H G is 1 cm, and the height is 4 cm. F E handan s Sample Papers : Sure Success and Excellent Marks 5

6 Find the length of [F]. Find the length of [F].(Total 8 marks) 12. In the diagram below EF, and FE are all rectangles. = 12 cm, = 20 cm and F = 5 cm. M is the midpoint of EF and N is the midpoint of. E M F 5 cm N 20 cm 12 cm alculate (i) the length of F; (ii) the length of M. alculate the angle between M and the face. (Total 6 marks) 13. gardener pegs out a rope, 19 metres long, to form a triangular flower bed as shown in this diagram. 5 m 6 m alculate the size of the angle ; the area of the flower bed. (Total 5 marks) 14. ndrew is at point in a park. deer is 3 km directly north of ndrew, at point. rian is 1.8 km due west of ndrew, at point. handan s Sample Papers : Sure Success and Excellent Marks 6

7 TRIN m raw a diagram to represent this information. alculate the distance between rian and the deer. rian looks at ndrew, and then turns through an angle θ to look at the deer. alculate the value of θ. 15. recreation park has two trains. Train 1 takes visitors from the entrance (E) to the swimming pool (S), to the mini golf (M) and back to the entrance. Train 2 takes visitors from the entrance (E) to the play area (P), to the racing track (R) and back to the entrance. This is shown in the diagram. (Total 8 marks) E TRIN m 115 S 400 m ES = 500 m SM = 400 m ER = 750 m ESM = 115 ERP = 50 EPR = 90 P [not to scale] M 50 R alculate the total distance Train 2 travels in one journey from E to P to R to E. (5) (i) Show that EM = 761 m correct to 3 s.f.. (ii) If the trains travel at 2 ms 1 find the time taken for Train 1 to complete a journey from E to S to M to E. Give your answer to the nearest second. (6) (Total 11 marks) 16. n office tower is in the shape of a cuboid with a square base. The roof of the tower is in the shape of a square based right pyramid. The diagram shows the tower and its roof with dimensions indicated. The diagram is not drawn O H 10 m G E F 40 m to scale. 6 m handan s Sample Papers : Sure Success and Excellent Marks 7

8 alculate, correct to three significant figures, (i) (ii) (iii) (iv) the size of the angle between OF and FG; the shortest distance from O to FG; the total surface area of the four triangular sections of the roof; the size of the angle between the slant height of the roof and the plane EFGH; (v) the height of the tower from the base to O. parrot's nest is perched at a point, P, on the edge, F, of the tower. person at the point, outside the building, measures the angle of elevation to point P to be 79. Find, correct to three significant figures, the height of the nest from the base of the tower. 17. The following diagram shows a triangle. = 8 m, = 14 m, = 18 m, and  = 110. (Total 14 marks) 110º 8 m 14 m 18 m alculate the area of triangle ; the size of angle Ĉ. 18. Three right pyramids ndal, atsu and artos were discovered in the dense jungle of Marhartmasol. Each pyramid has a square base with centres, and respectively. (Total 4 marks) handan s Sample Papers : Sure Success and Excellent Marks 8

9 ndal artos atsu surveying team was lowered from a helicopter to the top of ndal to take measurements of the area. ndal is 40 metres high. The angle of elevation from the top of ndal to the top of atsu is 3. The horizontal distance from, the centre of the base of ndal, to, the centre of the base of atsu is 600 metres. Use the diagram below to find the height of atsu. 3º 40 m ndal atsu 600 m artos is found to be 92 metres high and the angle of elevation from the top of ndal to the top of artos is 4. (i) raw a diagram similar to the diagram in part to show the relationship between ndal and artos. (ii) What is the horizontal distance from to? The diagram below represents measurements relative to the centres of the bases of the pyramids. The surveyors determined the angle at to be 110, and the distance to be 600 m. 110º (4) 600 m handan s Sample Papers : Sure Success and Excellent Marks 9

10 500 m (i) (ii) What is the distance between and? Give your answer to the nearest metre. What is the size of angle? (iii) What is the area of the land inside triangle? 19. is the point (2, 3), and is the point (4, 9). (8) (Total 15 marks) Find the gradient of the line segment []. Find the gradient of a line perpendicular to the line segment []. The line 2x + by 12 = 0 is perpendicular to the line segment []. What is the value of b? (Total 4 marks) 20. rectangular block of wood with face leans against a vertical wall, as shown in the diagram below. = 8 cm, = 5 cm and angle ÂE = 28. Wall F E 28º Find the vertical height of above the ground. Ground 21. cross-country running course is given in the diagram below. Runners start and finish at point O Not to scale (Total 4 marks) 1500 m 110 O. 800 m handan s Sample Papers : Sure Success and Excellent Marks 10

11 Show that the distance is 943 m correct to 3 s.f. Show that angle is 58.0 correct to 3 s.f. (i) alculate the angle O. (ii) alculate the distance O. (5) (d) (e) alculate the area enclosed by the course O. (4) Gonzales runs at a speed of 4 m s 1. alculate the time, in minutes, taken for him to complete the course. (Total 16 marks) 22. child's toy is made by combining a hemisphere of radius 3 cm and a right circular cone of slant height l as shown on the diagram below. diagram not to scale l 3 cm Show that the volume of the hemisphere is 18 cm 3. (d) (e) The volume of the cone is two-thirds that of the hemisphere. Show that the vertical height of the cone is 4 cm. alculate the slant height of the cone. alculate the angle between the slanting side of the cone and the flat surface of the hemisphere. The toy is made of wood of density 0.6 g per cm 3. alculate the weight of the toy. (4) (f) alculate the total surface area of the toy. (5) (Total 19 marks) handan s Sample Papers : Sure Success and Excellent Marks 11

12 23. farmer wants to construct a new fence across a field. The plan is shown below. The new fence is indicated by a dotted line m 40 alculate the length of the fence. (5) The fence creates two sections of land. Find the area of the smaller section of land, given the additional information shown below. 245 m 24 (Total 8 marks) 24. closed box has a square base of side x and height h. Write down an expression for the volume, V, of the box. Write down an expression for the total surface area,, of the box. (1) (1) The volume of the box is 1000 cm 3 Express h in terms of x. (d) Hence show that = 4000x l + 2x 2. (e) Find d. dx (f) alculate the value of x that gives a minimum surface area. (4) handan s Sample Papers : Sure Success and Excellent Marks 12

13 (g) Find the surface area for this value of x. (Total 15 marks) 25. The following diagram shows a carton in the shape of a cube 8 cm long on each side: F G H The longest rod that will fit on the bottom of the carton would go from E to G. Find the length l of this rod. Find the length L of the longest rod that would fit inside the carton. (Total 4 marks) handan s Sample Papers : Sure Success and Excellent Marks 13