Right Angled Trigonometry

1. A room is in the shape of a cuboid. Its floor measures 7.2 m by 9.6 m and itsheight is 3.5 m.

diagram not to scale

(a) Calculate the length of AC.(2)

(b) Calculate the length of AG.(2)

(c) Calculate the angle that AG makes with the floor.(2)

(Total 6 marks)

IB Questionbank Mathematical Studies 3rd edition 1

2. In the diagram, AD = 4 m, AB = 9 m, BC = 10 m, ADB = 90° and CBD = 100°.


(a) Calculate the size of CBA .(3)

(b) Calculate the length of AC.(3)

(Total 6 marks)


3. The diagram represents a small, triangular field, ABC, with BC = 25 m, angleBAC = 55° and angle ACB = 75°.


(a) Write down the size of angle ABC.(1)

(b) Calculate the length of AC.(3)

(c) Calculate the area of the field ABC.(3)

N is the point on AB such that CN is perpendicular to AB. M is the midpoint of CN.

(d) Calculate the length of NM.(3)


A goat is attached to one end of a rope of length 7 m. The other end of the rope is attached to the point M.

(e) Decide whether the goat can reach point P, the midpoint of CB. Justify your answer.

(5)(Total 15 marks)

4. José stands 1.38 kilometres from a vertical cliff.

(a) Express this distance in metres.(1)

José estimates the angle between the horizontal and the top of the cliff as 28.3° and uses it to find the height of the cliff.


(b) Find the height of the cliff according to José’s calculation. Express your answer in metres, to the nearest whole metre.

(3)

(c) The actual height of the cliff is 718 metres. Calculate the percentage error made by José when calculating the height of the cliff.

(2)(Total 6 marks)


5. A rectangular cuboid has the following dimensions.

Length 0.80 metres (AD)Width 0.50 metres (DG)Height 1.80 metres (DC)


(a) Calculate the length of AG.(2)

(b) Calculate the length of AF.(2)

(c) Find the size of the angle between AF and AG.(2)

(Total 6 marks)


6. The diagram shows triangle ABC. Point C has coordinates (4, 7) and the equation of the line AB is x + 2y = 8.


(a) Find the coordinates of

(i) A;

(ii) B.(2)

(b) Show that the distance between A and B is 8.94 correct to 3 significant figures.

(2)

N lies on the line AB. The line CN is perpendicular to the line AB.

(c) Find

(i) the gradient of CN ;

(ii) the equation of CN.(5)

(d) Calculate the coordinates of N.(3)


It is known that AC = 5 and BC = 8.06.

(e) Calculate the size of angle ACB.(3)

(f) Calculate the area of triangle ACB.(3)

(Total 18 marks)

7. The diagram shows an office tower of total height 126 metres. It consists of a square-based pyramid VABCD on top of a cuboid ABCDPQRS.

V is directly above the centre of the base of the office tower.

The length of the sloping edge VC is 22.5 metres and the angle that VC makes with the base ABCD (angle VCA) is 53.1°.


(a) (i) Write down the length of VA in metres.

(ii) Sketch the triangle VCA showing clearly the length of VC and the size of angle VCA.

(2)


(b) Show that the height of the pyramid is 18.0 metres correct to 3 significant figures.

(2)

(c) Calculate the length of AC in metres.(3)

(d) Show that the length of BC is 19.1 metres correct to 3 significant figures.

(2)

(e) Calculate the volume of the tower.(4)

To calculate the cost of air conditioning, engineers must estimate the weightof air in the tower. They estimate that 90 % of the volume of the tower is occupied by air and they know that 1 m3 of air weighs 1.2 kg.

(f) Calculate the weight of air in the tower.(3)

(Total 16 marks)

8. (a) A gardener has to pave a rectangular area 15.4 metres long and 5.5 metres wide using rectangular bricks. The bricks are 22 cm long and 11cm wide.

(i) Calculate the total area to be paved. Give your answer in cm2.

(ii) Write down the area of each brick.

(iii) Find how many bricks are required to pave the total area.(6)


(b) The gardener decides to have a triangular lawn ABC, instead of paving,in the middle of the rectangular area, as shown in the diagram below.


The distance AB is 4 metres, AC is 6 metres and angle BAC is 40°.

(i) Find the length of BC.

(ii) Hence write down the perimeter of the triangular lawn.

(iii) Calculate the area of the lawn.

(iv) Find the percentage of the rectangular area which is to be lawn.(9)


(c) In another garden, twelve of the same rectangular bricks are to be used to make an edge around a small garden bed as shown in the diagrams below. FH is the length of a brick and C is the centre of the garden bed. M and N are the midpoints of the long edges of the bricks on opposite sides of the garden bed.


(i) Find the angle FCH.

(ii) Calculate the distance MN from one side of the garden bed to the other, passing through C.

(5)

The garden bed has an area of 5419 cm2. It is covered with soil to a depth of2.5 cm.

(d) Find the volume of soil used.(2)

It is estimated that 1 kilogram of soil occupies 514 cm3.

(e) Find the number of kilograms of soil required for this garden bed.(2)

(Total 24 marks)


9. In the diagram below A , B and C represent three villages and the line segments AB, BC and CA represent the roads joining them. The lengths of AC and CB are 10 km and 8 km respectively and the size of the angle between them is 150°.


(a) Find the length of the road AB.(3)

(b) Find the size of the angle CAB.(3)

Village D is halfway between A and B. A new road perpendicular to AB and passing through D is built. Let T be the point where this road cuts AC. This information is shown in the diagram below.


(c) Write down the distance from A to D.(1)


(d) Show that the distance from D to T is 2.06 km correct to three significant figures.

(2)

A bus starts and ends its journey at A taking the route AD to DT to TA.

(e) Find the total distance for this journey.(3)

The average speed of the bus while it is moving on the road is 70 km h–1.The bus stops for 5 minutes at each of D and T.

(f) Estimate the time taken by the bus to complete its journey. Give your answer correct to the nearest minute.

(4)(Total 16 marks)

10. A and B are points on a straight line as shown on the graph below.

(a) Write down the y-intercept of the line AB.(1)


(b) Calculate the gradient of the line AB.(2)

The acute angle between the line AB and the x-axis is θ.

(c) Show θ on the diagram.(1)

(d) Calculate the size of θ.(2)

(Total 6 marks)

11. The right pyramid shown in the diagram has a square base with sides of length 40 cm.The height of the pyramid is also 40 cm.


(a) Find the length of OB.(4)

(b) Find the size of angle PBO .(2)

(Total 6 marks)


12. The diagram below shows a square based right pyramid. ABCD is a square of side 10 cm. VX is the perpendicular height of 8 cm. M is the midpoint of BC.


(a) Write down the length of XM.(1)

(b) Calculate the length of VM.(2)

(c) Calculate the angle between VM and ABCD.(2)

(Total 5 marks)


13. Triangle ABC is drawn such that angle CBA is 90, angle BCA is 60 and AB is 7.3 cm.

(a) (i) Sketch a diagram to illustrate this information. Label the points A, B, C.

Show the angles 90, 60 and the length 7.3 cm on your diagram.

(ii) Find the length of BC.(3)

Point D is on the straight line AC extended and is such that angle BDC is 20.

(b) (i) Show the point D and the angle 20 on your diagram.

(ii) Find the size of angle DBC .(3)

(Total 6 marks)


14. The triangular faces of a square based pyramid, ABCDE, are all inclined at 70 to the base. The edges of the base ABCD are all 10 cm and M is the centre. G is the mid-point of CD.

A

B

C

D

M

E

G

( D i a g r a m n o t t o s c a l e )

(a) Using the letters on the diagram draw a triangle showing the position of a 70 angle.

(1)

(b) Show that the height of the pyramid is 13.7 cm, to 3 significant figures.(2)

(c) Calculate

(i) the length of EG;

(ii) the size of angle CED .(4)

(d) Find the total surface area of the pyramid.(2)

(e) Find the volume of the pyramid.(2)

(Total 11 marks)


15. Sylvia is making a square-based pyramid. Each triangle has a base of length12 cm and a height of 10 cm.


(a) Show that the height of the pyramid is 8 cm.(2)

M is the midpoint of the base of one of the triangles and O is the apex of thepyramid.

(b) Find the angle that the line MO makes with the base of the pyramid.(3)

(c) Calculate the volume of the pyramid.(2)

(d) Daniel wants to make a rectangular prism with the same volume as that of Sylvia’s pyramid. The base of his prism is to be a square of side 10 cm. Calculate the height of the prism.

(2)(Total 9 marks)


16. The diagram shows a pyramid VABCD which has a square base of length 10 cm and edges of length 13 cm. M is the midpoint of the side BC.


(a) Calculate the length of VM.(2)

(b) Calculate the vertical height of the pyramid.(2)

(c) Calculate the angle between a sloping face of the pyramid and its base.

(2)(Total 6 marks)


17. An old tower (BT) leans at 10 away from the vertical (represented by line TG).

The base of the tower is at B so that TBM = 100.

Leonardo stands at L on flat ground 120 m away from B in the direction of the lean.

He measures the angle between the ground and the top of the tower T to beTLB = 26.5.

MM B = 2 0 0 B G B L = 1 2 0

2 6 . 5 °

n o t t o s c a l e

T

L9 0 °1 0 0 °

(a) (i) Find the value of angle LTB .

(ii) Use triangle BTL to calculate the sloping distance BT from the base, B to the top, T of the tower.

(5)

(b) Calculate the vertical height TG of the top of the tower.(2)

(c) Leonardo now walks to point M, a distance 200 m from B on the opposite side of the tower. Calculate the distance from M to the top of the tower at T.

(3)(Total 10 marks)


18. ABCDV is a solid glass pyramid. The base of the pyramid is a square of side 3.2 cm. The vertical height is 2.8 cm. The vertex V is directly above the centre O of the base.

V

A B

CD

O

(a) Calculate the volume of the pyramid.(2)

(b) The glass weighs 9.3 grams per cm3. Calculate the weight of the pyramid.

(2)

(c) Show that the length of the sloping edge VC of the pyramid is 3.6 cm.(4)

(d) Calculate the angle at the vertex, CVB .(3)

(e) Calculate the total surface area of the pyramid.(4)

(Total 15 marks)


19. Points P(0,–4), Q (0, 16) are shown on the diagram.

y

Q

0 x

P

2 4 6 8 1 0 1 2 1 4 1 6 1 8

8

(a) Plot the point R (11,16).

(b) Calculate angle R.PQ

M is a point on the line PR. M is 9 units from P.

(c) Calculate the area of triangle PQM.(Total 6 marks)


20. The figure below shows a rectangular prism with some side lengths and diagonal lengths marked. AC = 10 cm, CH = 10 cm, EH = 8cm, AE 8 cm.

A

B C

D

E

FG

H

1 0 c m

8 c m( n o t t o s c a l e )

1 0 c m

8 c m

(a) Calculate the length of AH.(2)

(b) Find the size of angle H.CA(3)

(c) Show that the total surface area of the rectangular prism is 320 cm2.(3)

(d) A triangular prism is enclosed within the planes ABCD, CGHD and ABGH. Calculate the volume of this prism.

(3)(Total 11 marks)

21. The diagram below shows a child’s toy which is made up of a circular hoop, centre O, radius 7 cm. The hoop is suspended in a horizontal plane by three equal strings XA, XB, and XC. Each string is of length 25 cm. The points A, B and C are equally spaced round the circumference of the hoop and X is vertically above the point O.



(a) Calculate the length of XO.(2)

(b) Find the angle, in degrees, between any string and the horizontal plane.

(2)

(c) Write down the size of angle B.OA(1)

(d) Calculate the length of AB.(3)

(e) Find the angle between strings XA and XB.(3)

(Total 11 marks)

22. The diagram shows a point P, 12.3 m from the base of a building of height h m. The angle measured to the top of the building from point P is 63°.

1 2 . 3

6 3 °P

h m

(a) Calculate the height h of the building.

Consider the formula h = 4.9t2, where h is the height of the building and t is the time in seconds to fall to the ground from the top of the building.

(b) Calculate how long it would take for a stone to fall from the top of the building to the ground.

(Total 6 marks)


23. A 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.


(a) Show that the volume of the hemisphere is 18p cm3.(2)

The volume of the cone is two-thirds that of the hemisphere.

(b) Show that the vertical height of the cone is 4 cm.(4)

(c) Calculate the slant height of the cone.(2)

(d) Calculate the angle between the slanting side of the cone and the flat surface of the hemisphere.

(3)

(e) The toy is made of wood of density 0.6 g per cm3. Calculate the weight of the toy.

(3)


(f) Calculate the total surface area of the toy.(5)

(Total 19 marks)

24. Find the volume of the following prism.

4 2 °

8 c m

5 . 7 c m

D i a g r a m n o t t o s c a l e

(Total 4 marks)


25.

A

B C

D

3 c m

4 . 5 c m

2 5 °

3 c m

In the diagram, AB = BC = 3 cm, DC = 4.5 cm, angle CBA = 90 and angle

DCA = 25.

(a) Calculate the length of AC.

(b) Calculate the area of triangle ACD.

(c) Calculate the area of quadrilateral ABCD.otal 8 marks)

26. The following diagram shows a sloping roof. The surface ABCD is a rectangle. The angle ADE is 55°. The vertical height, AF, of the roof is 3 m and the length DC is 7 m.

E F D

A

B

C

3 m7 m

5 5 °

(a) Calculate AD.


(b) Calculate the length of the diagonal DB.

(Total 8 marks)

27. OABCD is a square based pyramid of side 4 cm as shown in the diagram.The vertex D is 3 cm directly above X, the centre of square OABC.M is the midpoint of AB.

(a) Find the length of XM.

(b) Calculate the length of DM.

(c) Calculate the angle between the face ABD and the base OABC.

O

C

D

A

M

B

X

D i a g r a mn o t t o s c a l e

(Total 8 marks)


28. A cross-country running course is given in the diagram below. Runners start and finish at point O.

AB

C

O50

0 m

8 0 0 m

1 1 0 °

N o t t os c a l e

(a) Show that the distance CA is 943 m correct to 3 s.f.(2)

(b) Show that angle BCA is 58.0° correct to 3 s.f.(2)

(c) (i) Calculate the angle CAO.

(ii) Calculate the distance CO.(5)

(d) Calculate the area enclosed by the course OABC.(4)


(e) Gonzales runs at a speed of 4 m s–1. Calculate the time, in minutes, taken for him to complete the course.

(3)(Total 16 marks)

29. The figure below shows a hexagon with sides all of length 4 cm and with centre at O. The interior angles of the hexagon are all equal.

A

B

CD

E

F

O

4 c m

The interior angles of a polygon with n equal sides and n equal angles (regular polygon) add up to (n – 2) × 180°.

(a) Calculate the size of angle A B C.

(b) Given that OB = OC, find the area of the triangle OBC.

(c) Find the area of the whole hexagon.

(Total 8 marks)


30. A 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 racingtrack (R) and back to the entrance. This is shown in the diagram.

1 1 5 °

5 0 °

TR

AIN

2

ES

M

P

R

750

m

4 0 0 m

E S = 5 0 0 mS M = 4 0 0 mE R = 7 5 0 mE S M = 1 1 5 °E R P = 5 0 °E P R = 9 0 °

[ n o t t o s c a l e ]

(a) Calculate the total distance Train 2 travels in one journey from E to P to R to E.

(5)

(b) (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)


31. In the diagram below ABEF, ABCD and CDFE are all rectangles. AD = 12 cm, DC = 20 cm and DF = 5 cm.M is the midpoint of EF and N is the midpoint of CD.

A

B C

D

E

F

1 2 c m

5 c m

M

N

(a) Calculate (i) the length of AF;

(ii) the length of AM.(3)

(b) Calculate the angle between AM and the face ABCD.(3)

(Total 6 marks)

32. Andrew is at point A in a park. A deer is 3 km directly north of Andrew, at point D. Brian is 1.8 km due west of Andrew, at point B.

(a) Draw a diagram to represent this information.

(b) Calculate the distance between Brian and the deer.

(c) Brian looks at Andrew, and then turns through an angle θ to look at thedeer. Calculate the value of θ.

(Total 8 marks)


33. Three right pyramids Andal, Batsu and Cartos were discovered in the dense jungle of Marhartmasol. Each pyramid has a square base with centres A, B and C respectively.

A

B

C

C a r t o s

B a t s u

A n d a l

Diagram not to scale

A surveying team was lowered from a helicopter to the top of Andal to take measurements of the area. Andal is 40 metres high. The angle of elevation from the top of Andal to the top of Batsu is 3°. The horizontal distance from A, the centre of the base of Andal, to B, the centre of the base of Batsu is 600 metres.

(a) Use the diagram below to find the height of Batsu.(3)


3 º

A n d a l B a t s u

A B

6 0 0 m

4 0 m


(b) Cartos is found to be 92 metres high and the angle of elevation from the top of Andal to the top of Cartos is 4°.

(i) Draw a diagram similar to the diagram in part (a) to show the relationship between Andal and Cartos.

(ii) What is the horizontal distance from A to C?(4)

(c) The diagram below represents measurements relative to the centres ofthe bases of the pyramids. The surveyors determined the angle at A to be 110°, and the distance AB to be 600 m.


A

B

C

6 0 0 m

1 1 0 º

(i) What is the distance between B and C? Give your answer to the nearest metre.

(ii) What is the size of angle ACB?

(iii) What is the area of the land inside triangle ABC?(8)

(Total 15 marks)


34. The following diagram shows the rectangular prism ABCDEFGH. The length is 5 cm, the width is 1 cm, and the height is 4 cm.

C H

BG

DE

A F Diagram not to scale

(a) Find the length of [DF].

(b) Find the length of [CF].

(Total 8 marks)

35. The diagram below represents a stopwatch. This is a circle, centre O, inside a square of side 6 cm, also with centre O. The stopwatch has a minutes hand and a seconds hand. The seconds hand, with end point T, is shown in the diagram, and has a radius of 2 cm.

A

DT

O

B

C

6 c m

r

qp

(a) When T is at the point A, the shortest distance from T to the base of the square is p. Calculate the value of p.

(2)


(b) In 10 seconds, T moves from point A to point B. When T is at the point B, the shortest distance from T to the base of the square is q. Calculate

(i) the size of angle AOB;

(ii) the distance OD;

(iii) the value of q.(5)

(c) In another 10 seconds, T moves from point B to point C. When T is at the point C, the shortest distance from T to the base of the square is r. Calculate the value of r.

(4)

Let d be the shortest distance from T to the base of the square, when the seconds hand has moved through an angle q. The following table gives values of d and q.

Angle 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330 360°

q

Distance p 4.7 q 3 r 1.3 1 1.3 r 3 q 4.7 p

d


The graph representing this information is as follows.

d

p

q

r

0 º 3 0 º 6 0 º 9 0 º 1 2 0 º 1 5 0 º 1 8 0 º 2 1 0 º 2 4 0 º 2 7 0 º 3 0 0 º 3 3 0 º 3 6 0 ºa n g l e

The equation of this graph can be written in the form d = c + k cos(q).

(d) Find the values of c and k.(4)

(Total 15 marks)

36. A rectangular block of wood with face ABCD leans against a vertical wall, as

shown in the diagram below. AB = 8 cm, BC = 5 cm and angle EAB = 28°.

W a l lF

B

E

C

D

A

G r o u n d2 8 º

Find the vertical height of C above the ground.

(Total 4 marks)


37. ABCD is a trapezium with AB = CD and [BC] parallel to [AD]. AD = 22 cm, BC = 12 cm, AB = 13 cm.


B C

A E D

(a) Show that AE = 5 cm.(2)

(b) Calculate the height BE of the trapezium.(2)

(c) Calculate

(i) E;AB

(ii) D.CB(3)

(d) Calculate the length of the diagonal [CA].(3)

(Total 10 marks)


38. The diagram shows a cuboid 22.5 cm by 40 cm by 30 cm.

H G

E F

A B

D C

4 0 c m

3 0 c m

2 2 . 5 c m

(a) Calculate the length of [AC].

(b) Calculate the size of CAG .

(Total 4 marks)

39. 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

P

G

Q

M

RS

V G = 1 2 c m

8 c m

8 c m

(a) (i) Write down the length of [GM].

(ii) Calculate the length of [VM].(2)

(b) Find


(i) the total surface area of the pyramid;

(ii) the angle between the face VQR and the base of the pyramid.(4)

(Total 6 marks)

40. The following diagram shows a carton in the shape of a cube 8 cm long on each side:

B C

E H

AD

GF

(a) 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.

(b) Find the length L of the longest rod that would fit inside the carton.

(Total 4 marks)

41. The height of a vertical cliff is 450 m. The angle of elevation from a ship to the top of the cliff is 23°. The ship is x metres from the bottom of the cliff.

(a) Draw a diagram to show this information.

Diagram:

(b) Calculate the value of x.


(Total 4 marks)

42. The diagram shows a water tower standing on horizontal ground. The heightof the tower is 26.5 m.

Ax m

From a point A on the ground the angle of elevation to the top of the tower is 28°.

(a) On the diagram, show and label the angle of elevation, 28°.

(b) Calculate, correct to the nearest metre, the distance x m.

(Total 4 marks)


Right Angled Trigonometry

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