5.2

1a. [3 marks]

A flat horizontal area, ABC, is such that AB = 100 m , BC = 50 m and angle ACB = 43.7° as shown in the

diagram.

Show that the size of angle BAC is 20.2°, correct to 3 significant figures.

1b. [4 marks]

Calculate the area of triangle ABC.

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1c. [3 marks]

Find the length of AC.

1d. [5 marks]

A vertical pole, TB, is constructed at point B and has height 25 m.

Calculate the angle of elevation of T from, M, the midpoint of the side AC.

 

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2a. [3 marks]

Farmer Brown has built a new barn, on horizontal ground, on his farm. The barn has a cuboid base and

a triangular prism roof, as shown in the diagram.

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The cuboid has a width of 10 m, a length of 16 m and a height of 5 m.

The roof has two sloping faces and two vertical and identical sides, ADE and GLF.

The face DEFL slopes at an angle of 15° to the horizontal and ED = 7 m .

Calculate the area of triangle EAD.

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2b. [3 marks]

Calculate the total volume of the barn.

2c. [2 marks]

The roof was built using metal supports. Each support is made from five lengths of metal AE, ED, AD,

EM and MN, and the design is shown in the following diagram.

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ED = 7 m , AD = 10 m and angle ADE = 15° .

M is the midpoint of AD.

N is the point on ED such that MN is at right angles to ED.

Calculate the length of MN.

2d. [3 marks]

Calculate the length of AE.

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2e. [3 marks]

Farmer Brown believes that N is the midpoint of ED.

Show that Farmer Brown is incorrect.

2f. [4 marks]

Calculate the total length of metal required for one support.

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3a. [5 marks]

The Tower of Pisa is well known worldwide for how it leans.

Giovanni visits the Tower and wants to investigate how much it is leaning. He draws a diagram showing

a non-right triangle, ABC.

On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and angle ACB is 60°. AX is the

perpendicular height from A to BC.

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Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the

horizontal, is 85° to the nearest degree.

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3b. [2 marks]

Use Giovanni's diagram to calculate the length of AX.

3c. [2 marks]

Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.

3d. [2 marks]

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Giovanni’s tourist guidebook says that the actual horizontal displacement of the Tower, BX, is 3.9

metres.

Find the percentage error on Giovanni’s diagram.

3e. [3 marks]

Giovanni adds a point D to his diagram, such that BD = 45 m, and another triangle is formed.

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Find the angle of elevation of A from D.

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4a. [3 marks]

Emily’s kite ABCD is hanging in a tree. The plane ABCDE is vertical.

Emily stands at point E at some distance from the tree, such that EAD is a straight line and angle BED =

7°. Emily knows BD = 1.2 metres and angle BDA = 53°, as shown in the diagram

Find the length of EB.

4b. [1 mark]

T is a point at the base of the tree. ET is a horizontal line. The angle of elevation of A from E is 41°.13

Write down the angle of elevation of B from E.

4c. [2 marks]

Find the vertical height of B above the ground.

5a. [2 marks]

A lampshade, in the shape of a cone, has a wireframe consisting of a circular ring and four straight

pieces of equal length, attached to the ring at points A, B, C and D.

The ring has its centre at point O and its radius is 20 centimetres. The straight pieces meet at point V,

which is vertically above O, and the angle they make with the base of the lampshade is 60°.

This information is shown in the following diagram.

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Find the length of one of the straight pieces in the wireframe.

5b. [4 marks]

Find the total length of wire needed to construct this wireframe. Give your answer in centimetres

correct to the nearest millimetre.

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6a. [2 marks]

The base of an electric iron can be modelled as a pentagon ABCDE, where:

Write down an equation for the area of ABCDE using the above information.

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6b. [2 marks]

Show that the equation in part (a)(i) simplifies to .

6c. [2 marks]

Find the length of CD.

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6d. [3 marks]

Show that angle , correct to one decimal place.

6e. [3 marks]

Insulation tape is wrapped around the perimeter of the base of the iron, ABCDE.

Find the length of the perimeter of ABCDE.

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6f. [4 marks]

F is the point on AB such that . A heating element in the iron runs in a straight line, from C

to F.

Calculate the length of CF.

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7a. [2 marks]

AC is a vertical communications tower with its base at C.

The tower has an observation deck, D, three quarters of the way to the top of the tower, A.

From a point B, on horizontal ground 250 m from C, the angle of elevation of D is 48°.

Calculate CD, the height of the observation deck above the ground.

7b. [4 marks]

Calculate the angle of depression from A to B.

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8a. [2 marks]

Temi’s sailing boat has a sail in the shape of a right-angled triangle, ,

angle and angle .

Calculate , the height of Temi’s sail.

8b. [2 marks]

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William also has a sailing boat with a sail in the shape of a right-angled triangle, .

. The area of William’s sail is .

Calculate , the height of William’s sail.

9a. [2 marks]

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The equation of line is . Point lies on .

Find the value of .

9b. [1 mark]

The line is perpendicular to and intersects at point .

Write down the gradient of .

9c. [2 marks]

Find the equation of . Give your answer in the form .

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9d. [1 mark]

Write your answer to part (c) in the form  where , and .

10a. [1 mark]

A ladder is standing on horizontal ground and leaning against a vertical wall. The length of the ladder is

metres. The distance between the bottom of the ladder and the base of the wall is metres.

Use the above information to sketch a labelled diagram showing the ground, the ladder and the wall.

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10b. [2 marks]

Calculate the distance between the top of the ladder and the base of the wall.

10c. [3 marks]

Calculate the obtuse angle made by the ladder with the ground.

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11a. [3 marks]

The Great Pyramid of Giza in Egypt is a right pyramid with a square base. The pyramid is made of solid

stone. The sides of the base are long. The diagram below represents this pyramid, labelled

.

is the vertex of the pyramid.  is the centre of the base, . is the midpoint of . Angle

.

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Show that the length of is metres, correct to three significant figures.

11b. [2 marks]

Calculate the height of the pyramid, .

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11c. [2 marks]

Find the volume of the pyramid.

11d. [2 marks]

Write down your answer to part (c) in the form  where and .

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11e. [4 marks]

Ahmad is a tour guide at the Great Pyramid of Giza. He claims that the amount of stone used to build the

pyramid could build a wall metres high and  metre wide stretching from Paris to Amsterdam, which

are apart.

Determine whether Ahmad’s claim is correct. Give a reason.

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11f. [6 marks]

Ahmad and his friends like to sit in the pyramid’s shadow, , to cool down.

At mid-afternoon,  and angle 

i)     Calculate the length of  at mid-afternoon.

ii)    Calculate the area of the shadow, , at mid-afternoon.

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12a. [3 marks]

A playground, when viewed from above, is shaped like a quadrilateral, , where   

and . Three of the internal angles have been measured and angle , angle

and angle . This information is represented in the following diagram.

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Calculate the distance .

12b. [3 marks]

Calculate angle .

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12c. [2 marks]

There is a tree at , perpendicular to the ground. The angle of elevation to the top of the tree from  is

.

Calculate the height of the tree.

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12d. [2 marks]

Chavi estimates that the height of the tree is .

Calculate the percentage error in Chavi’s estimate.

12e. [3 marks]

Chavi is celebrating her birthday with her friends on the playground. Her mother brings a bottle

of orange juice to share among them. She also brings cone-shaped paper cups.

Each cup has a vertical height of and the top of the cup has a diameter of .

Calculate the volume of one paper cup.

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12f. [3 marks]

Calculate the maximum number of cups that can be completely filled with the bottle of orange

juice.

 

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13a. [1 mark]

In the following diagram, ABCD is the square base of a right pyramid with vertex V. The centre of the

base is O. The diagonal of the base, AC, is 8 cm long. The sloping edges are 10 cm long.

Write down the length of .

13b. [2 marks]

Find the size of the angle that the sloping edge  makes with the base of the pyramid.

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13c. [3 marks]

Hence, or otherwise, find the area of the triangle .

14a. [2 marks]

Fabián stands on top of a building, T, which is on a horizontal street.

He observes a car, C, on the street, at an angle of depression of 30°. The base of the building is at B. The

height of the building is 80 metres.

The following diagram indicates the positions of T, B and C.

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Show, in the appropriate place on the diagram, the values of

(i)     the height of the building;

(ii)     the angle of depression.

14b. [2 marks]

Find the distance, BC, from the base of the building to the car.

14c. [2 marks]

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Fabián estimates that the distance from the base of the building to the car is 150 metres. Calculate the

percentage error of Fabián’s estimate.

15a. [4 marks]

A boat race takes place around a triangular course, , with ,  and angle

. The race starts and finishes at point .

Calculate the total length of the course.

15b. [3 marks]

It is estimated that the fastest boat in the race can travel at an average speed of .

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Calculate an estimate of the winning time of the race. Give your answer to the nearest minute.

15c. [3 marks]

It is estimated that the fastest boat in the race can travel at an average speed of .

Find the size of angle .

15d. [3 marks]

To comply with safety regulations, the area inside the triangular course must be kept clear of other

boats, and the shortest distance from  to  must be greater than  metres.

Calculate the area that must be kept clear of boats.

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15e. [3 marks]

To comply with safety regulations, the area inside the triangular course must be kept clear of other

boats, and the shortest distance from  to  must be greater than  metres.

Determine, giving a reason, whether the course complies with the safety regulations.

15f. [2 marks]

The race is filmed from a helicopter, , which is flying vertically above point .

The angle of elevation of  from  is .

Calculate the vertical height, , of the helicopter above .

15g. [3 marks]

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The race is filmed from a helicopter, , which is flying vertically above point .

The angle of elevation of  from  is .

Calculate the maximum possible distance from the helicopter to a boat on the course.

16a. [2 marks]

A child’s wooden toy consists of a hemisphere, of radius 9 cm , attached to a cone with the same base

radius. O is the centre of the base of the cone and V is vertically above O.

Angle OVB is .

Diagram not to scale.

Calculate OV, the height of the cone.

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16b. [4 marks]

Calculate the volume of wood used to make the toy.

17a. [3 marks]

Günter is at Berlin Tegel Airport watching the planes take off. He observes a plane that is at an angle of

elevation of from where he is standing at point . The plane is at a height of 350 metres. This

information is shown in the following diagram.

Calculate the horizontal distance, , of the plane from Günter. Give your answer to the nearest

metre.

17b. [3 marks]

The plane took off from a point , which is metres from where Günter is standing, as shown in the

following diagram.

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Using your answer from part (a), calculate the angle , the takeoff angle of the plane.

18a. [3 marks]

ABC is a triangular field on horizontal ground. The lengths of AB and AC are 70 m and 50 m

respectively. The size of angle BCA is 78°.

Find the size of angle .

18b. [4 marks]

Find the area of the triangular field.

18c. [3 marks]

is the midpoint of  .

Find the length of  .

18d. [5 marks]

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A vertical mobile phone mast, , is built next to the field with its base at  . The angle of elevation

of   from   is  .   is the midpoint of the mast.

 

Calculate the angle of elevation of   from  .

19a. [2 marks]

In triangle , , and .

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     diagram not to scale

Find the length of .

19b. [2 marks]

is the point on   such that .

Find the length of  .

19c. [2 marks]

 is the point on   such that .

Find the area of triangle  .

20a. [3 marks]

A tent is in the shape of a triangular right prism as shown in the diagram below.

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The tent has a rectangular base PQRS .

PTS and QVR are isosceles triangles such that PT = TS and QV = VR .

PS is 3.2 m , SR is 4.7 m and the angle TSP is 35°.

Show that the length of side ST is 1.95 m, correct to 3 significant figures.

20b. [3 marks]

Calculate the area of the triangle PTS.

20c. [1 mark]

Write down the area of the rectangle STVR.

20d. [3 marks]

Calculate the total surface area of the tent, including the base.

20e. [2 marks]

Calculate the volume of the tent.

20f. [4 marks]

A pole is placed from V to M, the midpoint of PS.

Find in metres,

(i) the height of the tent, TM;

(ii) the length of the pole, VM.

20g. [2 marks] 47

Calculate the angle between VM and the base of the tent.

21a. [2 marks]

A solid metal cylinder has a base radius of 4 cm and a height of 8 cm.

Find the area of the base of the cylinder.

21b. [2 marks]

Show that the volume of the metal used in the cylinder is 402 cm3, given correct to three significant

figures.

21c. [3 marks]

Find the total surface area of the cylinder.

21d. [3 marks]

The cylinder was melted and recast into a solid cone, shown in the following diagram. The base radius

OB is 6 cm.

Find the height, OC, of the cone.

 

21e. [2 marks]

The cylinder was melted and recast into a solid cone, shown in the following diagram. The base radius

OB is 6 cm.

48

Find the size of angle BCO.

21f. [2 marks]

The cylinder was melted and recast into a solid cone, shown in the following diagram. The base radius

OB is 6 cm.

Find the slant height, CB.

21g. [4 marks]

The cylinder was melted and recast into a solid cone, shown in the following diagram. The base radius

OB is 6 cm.

49

Find the total surface area of the cone.

22a. [3 marks]

The Great Pyramid of Cheops in Egypt is a square based pyramid. The base of the pyramid is a square of

side length 230.4 m and the vertical height is 146.5 m. The Great Pyramid is represented in the diagram

below as ABCDV . The vertex V is directly above the centre O of the base. M is the midpoint of BC.

(i) Write down the length of OM .

(ii) Find the length of VM .

22b. [2 marks]

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Find the area of triangle VBC .

22c. [2 marks]

Calculate the volume of the pyramid.

22d. [2 marks]

Show that the angle between the line VM and the base of the pyramid is 52° correct to 2 significant

figures.

22e. [1 mark]

Ahmed is at point P , a distance x metres from M on horizontal ground, as shown in the following

diagram. The size of angle VPM is 27° . Q is a point on MP .

Write down the size of angle VMP .

22f. [4 marks]

Ahmed is at point P , a distance x metres from M on horizontal ground, as shown in the following

diagram. The size of angle VPM is 27° . Q is a point on MP .

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Using your value of VM from part (a)(ii), find the value of x.

22g. [4 marks]

Ahmed is at point P , a distance x metres from M on horizontal ground, as shown in the following

diagram. The size of angle VPM is 27° . Q is a point on MP .

Ahmed walks 50 m from P to Q.

Find the length of QV, the distance from Ahmed to the vertex of the pyramid.

23a. [3 marks] 52

In the diagram, , , , and .

Calculate the size of .

23b. [3 marks]

Calculate the length of AC.

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