DP IB Maths: AA SL

Topic Questions

3.3 Trigonometry

1a
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3 marks

Adah would like to estimate the height of a tree located at point P on the edge of a riverbank, with the top of the tree at point Q. However, due to a raging river, she is unable to reach the base of the tree. From point M she measures an angle of elevation of 20° to the top of the tree, and then from point N (which is on the edge of Adah’s bank of the river) she measures an angle of elevation of 35° to the top of the tree. Between the points M and N she measures a horizontal distance of 12 m. Points M, N and P all lie on a single horizontal line, and point Q is vertically above point P. The diagram below shows this information.

q1-3-3-hard-trigonometry-ib-maths-

Calculate the length of NQ.

1b
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2 marks

Calculate the height of the tree. 

1c
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3 marks

Adah borrows a boat and crosses the river at a rate of 50 metres per 15 minutes. 

Assuming that she crosses in a straight line directly from point N to point P, find out how long it takes her to cross the river.

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2a
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3 marks

The diagram below shows a triangular field on a farm. AB = 17 m, AC = 45 m and angle straight B straight A with hat on top straight C = 38°.

X is a point on AC, such that AX :XC is 1 :4.

q2-3-3-hard-trigonometry-ib-maths-

The field is going to be used for livestock, so a fence is to be installed around its perimeter.

Calculate the total length of fencing required.

2b
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4 marks

The field is to be divided into two parts by installing a new fence connecting B to X.

Calculate the area of BXC.

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3a
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3 marks

The diagram below shows a ship that is located 86 m from an observation station, and a whale that has been spotted from the observation station at a distance of 137 m.

q3a-3-3-hard-ib-aa-sl-maths

The bearing of the ship from the observation station is 110 degree and the whale is located along a bearing of 072 degree from the observation station. 

Calculate the distance between the ship and the whale.

3b
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4 marks

Work out the bearing that the ship needs to travel on to reach the whale. Give your answer to 1 decimal place.

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4a
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3 marks

The cross-section of a unicorn horn can be modelled by the triangle ABC shown in the diagram below. The length AB = 49 cm and length BC = 58 cm.  The cross-sectional area of the horn is 168 cm2.

q3-3-3-hard-trigonometry-ib-maths-

Find the size of the angle straight A straight B with hat on top straight C formed at the tip of horn.

4b
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3 marks

Calculate the length of the base AC that is attached to the unicorn’s head.

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5a
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2 marks

The diagram below shows a quadrilateral ABCD.  Angle straight B straight A with hat on top straight D = 59 degree and angle straight B straight C with hat on top straight D = 46 degree. AB = 14.4 cm, AD = 16.2 cm and BC = 19.7 cm.

q4-3-3-hard-trigonometry-ib-maths-

Calculate the length BD.

5b
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2 marks

Find the size of the angle straight C straight D with hat on top straight B.

5c
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2 marks

Show that the area of the quadrilateral is 235 cm2 correct to the nearest cm2.

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6a
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4 marks

The diagram below shows the triangular sail of a windsurfing board, ABC, with a horizontal boom PC. AB = 6.1 m and makes an angle of 18° to the vertical. BC = 4.7 m and straight B straight C with hat on top straight P = 70°.

q5-3-3-hard-trigonometry-ib-maths-

Find the area of the whole sail.

6b
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2 marks

Calculate the length of the boom PC.

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7a
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5 marks

The area of triangle ABC is 12 square root of 2.

q6-3-3-hard-trigonometry-ib-maths-

Calculate the value of x.

7b
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3 marks

Hence, find BC.

7c
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3 marks

Heron’s formula states that it is actually possible to find the area of any triangle given only its side lengths a comma space b and c.

Area space equals space square root of s open parentheses s minus a close parentheses open parentheses s minus b close parentheses open parentheses s minus c close parentheses end root

The value of s is known as the ‘half perimeter’ where s space equals space fraction numerator a plus b plus c over denominator 2 end fraction

Verify that Heron’s formula works for triangle ABC.

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8a
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5 marks

The diagram shows a triangular prism ABCDEF of height 18.2 cm.  ED = 7.5 cm, EF = 5.3 cm and AC = 6.6 cm. 

q7-3-3-hard-trigonometry-ib-maths-

M is the midpoint of BC.

Calculate the length DM.

8b
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3 marks

Find the size of the angle straight E straight M with hat on top straight D.

8c
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2 marks

Find the area of the triangle EDM.

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9a
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4 marks

The diagram below shows a cable-stayed bridge crossing a river from A to B. The height of the embankment at A, measured from the horizontal river bed, is 9.1 m and this drops to a height of 1.3 m at B. The width of the river bed is 90 m.  A vertical central column of height 15 m is situated at the midpoint of the river bed, P, and connects to the exterior supporting cables at point V. The other ends of the cables are attached at points A and B respectively.

q8-3-3-hard-trigonometry-ib-maths-

Find the size of angle straight V straight B with hat on top straight A, between the exterior supporting cable and the bridge span.

9b
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6 marks

Calculate the total length of the two exterior supporting cables.

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1a
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4 marks

Owen, Henry and Tom are rugby players passing a ball in a park. Owen is at point straight O, Henry is at point straight H and Tom is at point straight T. The distance between Owen and Henry is 25 m and the distance between Henry and Tom is 18 m. The angle straight O straight H with hat on top straight T is 96 degree.

(i)
Draw and label a diagram to represent the situation described above.

(ii)
Find the length of the line OT.
1b
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3 marks

Find the size of the angle straight O straight T with hat on top straight H.

1c
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3 marks

Find the area of the section of the park the players are using to pass the ball.

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2a
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3 marks

straight A sailboat race takes place annually for under 18’s on a large lake. The competitors must sail around five flagged buoys at the points straight A comma space straight B comma space straight C comma space straight D and straight E, in a clockwise direction.

straight B is due east of straight A comma space straight C is due south of straight B and straight A is due north of straight E
The bearing from straight A to straight C is 110° and the bearing from straight C to straight D is 220°.
The distance AB = 1200 m, the distance BC = 600 m, the distance CD = 800 m and the distances DE = EA = 1000 m.

Draw and label a diagram to show the buoys A, B, C, D and E and clearly mark the bearings and distances given above.

2b
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2 marks

The boats all start at A and must complete the course 5 times. A support motorboat is present and can travel across the course from A to C and A to D in case of an emergency.

Calculate the distance from A to C.

2c
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3 marks

Calculate the distance from A to D.

2d
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4 marks

Calculate the bearing the support boat must follow to travel from A to D.

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3a
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2 marks

The following diagram shows triangle ABC. AC = 21 km, CB = 15 km, straight A straight C with hat on top straight B = 75°.

q3-3-3-medium-trigonometry-ib-maths-

Find the area of triangle ABC.

3b
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3 marks

Find AB.

3c
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2 marks

Given that it is acute, find straight C straight A with hat on top straight B.

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4a
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1 mark

Triangle ABC has an area of 122 cm2, AB = 24 cm and BC = 11 cm.

Draw and label a diagram to show triangle ABC and clearly mark the distances given.

4b
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4 marks

Given that straight A straight B with hat on top straight C is acute, find

(i)
straight A straight B with hat on top straight C

(ii)
AC.

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5a
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3 marks

The quadrilateral ABCD shown below represents a farm paddock, where AB = 246 m, BC = 312 m and AD = 257 m. Angle straight D straight A with hat on top straight B=96° and angle straight B straight C with hat on top straight D = 78°.

q5-3-3-medium-trigonometry-ib-maths-

A fence is built connecting points B and D to split the paddock into two.

Find the length of the fence.

5b
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5 marks

Find the area of the paddock ABCD.

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6a
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2 marks

A 38 m high cliff is perpendicular to the sea and the angle of depression from the cliff to a boat at sea is 24°. Climbing the cliff is a rock climber and the angle of elevation from the boat to the climber 14°.

Draw and label a diagram to show the top of the cliff, T, the foot of the cliff, F, the climber, C, the boat, B, labelling all the angles and distances given above.

6b
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2 marks

Find the distance from the boat to the foot of the cliff.

6c
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4 marks

Find how far the climber must climb to reach the top of the cliff.

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7a
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2 marks

The diagram below shows triangle XYZ with side length YZ = 5.4 cm. The point W is placed such that XW = 5.6 cm and WZ = 4.2 cm and YW = 5.8 cm.

q7-3-3-medium-trigonometry-ib-maths-

Find the angle straight Y straight Z with hat on top straight W.

7b
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2 marks

Find the area of triangle XYZ.

7c
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3 marks

Find the area of triangle XYW.

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8a
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2 marks

The distance between towns X and Y is 134.2 km. The bearing of town X from town Y is 119°. Town Z is 54 km south of town X. The bearing of town Z from town X is 207°.

Draw and label a diagram to show towns X,Y and Z, clearly marking the bearings and distances given above.

8b
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2 marks

Calculate the distance between towns X and Z.

8c
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4 marks

Calculate the distance between towns Y and Z.

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9
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8 marks

The diagram below shows four Islands P,Q,R and S. PQ = 8.5 km, QR = 16.2 km and RS = 12.5 km. Angle straight P straight Q with hat on top straight S=25°, angle straight Q straight S with hat on top straight P = 14.7° and angle straight Q straight R with hat on top straight S = 82.1°. Island Q is due north from Island P.

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Mark is making deliveries around the Islands. He takes milk from Island Q to Island S, then takes wood from Island S to Island P, finally he delivers fruit from Island P to Island R.

Find the total distance Mark travels.

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10a
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2 marks

Nathan (N)  stands 10 m above the ground on the second-floor balcony of an apartment building and can see Melissa (M) in the car park. The angle of elevation from Melissa to Nathan is 21.6°.

Calculate the distance from M to N.

10b
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3 marks

Louisa (L) is standing on the other side of the car park. The distance between Louisa and Nathan is 1.5 times the distance between Melissa and Nathan.

Calculate the angle of depression from N to L.

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1a
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3 marks

The diagram below shows a rectangular based pyramid ABCDE. DC = 5.9 cm, AD = 3.7 cm and AE = 7.4 cm. The vertex, E, is positioned directly above the midpoint of the base ABCD.

q1-3-3-very-hard-trigonometry-ib-maths-

Find the size of angle straight A straight E with hat on top straight C.

1b
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3 marks

P is a point located on the edge EB, such that EP : PB = 1 : 4.

Find the area of the triangle EPD.

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2a
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3 marks

The diagram below shows a police helicopter using a high beam light at point B to search an area on the ground between A and C. The length of the edge of the light beam that is furthest from the helicopter is 22 m and the angle of depression from the helicopter to the same edge is 47degree.

q2-3-3-very-hard-trigonometry-ib-maths-

The area of the cross section of the search beam, ABC, is 23 m2.

Calculate the horizontal length of ground, AC, that is lit by the beam.

2b
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4 marks

Find the size of the angle of the beam straight A straight B with hat on top straight C.

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3
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7 marks

A spider starts to weave a web, ABCDEFGO, with threads of equal length (AB, BC, etc.) linking the 7 vertices that are equally spaced around the centre point, O. Threads connecting each vertex to the centre (OA, OB, etc.) are also created by the spider. Each line from the centre has a length of 12.6 cm, and the points O, A, B, C, D, E, F and G all lie in a single plane. This is represented in the diagram below.

q3-3-3-very-hard-trigonometry-ib-maths-

The spider decides to add more silk to its web by connecting each vertex to the midpoint of the adjacent line when moving clockwise around the web, for example from G to the midpoint of OA.

Given that the spider can produce 220 cm of silk in a day, find the maximum possible length for thread OA if the spider is to be able to complete the web in one day.

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4a
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6 marks

The diagram below shows a funnel in the shape of a right cone with a smaller cone removed from the end. The circular planes at both ends of the object are parallel to one another. The perpendicular height of the complete cone is 168 mm. The initial diameter of the funnel is 98 mm at the upper end and narrows to a diameter of 7 mm at the bottom. 

The funnel has been used to pour some sugar into a bottle and a grain of sugar remains at point P, 1 third of the way up the slanted height of the funnel. An ant sits on the edge at the top of the funnel at point Aopen square brackets AB close square brackets is a diameter of the large circular face. Points A, B, P and the axis of the cone all lie on a single plane.

q4-3-3-very-hard-trigonometry-ib-maths-

Calculate the direct distance between the ant and the grain of sugar, AP.

4b
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3 marks

Find the size of the angle of depression from the ant to the grain of sugar.

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5a
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4 marks

A piece of equipment in a playground consists of two ropes fixed to a hook at point A at the edge of a gap and pulled taut to the other side and fixed at points B and C. The height of the taller left embankment above the ground is 2.3 m. Point B is located at the top of the right embankment. Point C is vertically beneath point B, and the height from point C to the ground is 0.8 m. The angle between the ropes is 23degree and the horizontal distance of the gap that is being bridged is 1.4 m. The information is shown on the diagram below. 

q5-3-3-very-hard-trigonometry-ib-maths-

Calculate the length BC.

5b
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4 marks

A third piece of rope of length 0.9 m is to be added to the structure at point B and fixed at a point P on the rope AC

Calculate the angle straight B straight P with hat on top straight C and hence the distance PC.

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6
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4 marks

A symmetrical pendant for a necklace is made in the shape of an irregular hexagon, ABCDEF, as can be seen in the diagram below.  AF, BE and CD are parallel. BE = 22 mm,  AF = CD = 16 mmand text A end text stack text B end text with hat on top text C end text = 100degree. AB = BC and EF = DE. The shaded area is silver and the white area is gold. The width of the entire pendant is 54 mm. This information is shown in the diagram below:

q6-3-3-very-hard-trigonometry-ib-maths-

Calculate the percentage of the area of the pendant that is silver.

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7a
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2 marks

The ‘H’ on the Hollywood sign is 13.7 m tall, with each leg being 3 m wide and the gap between the legs being 4 m. The width of the cross bar of the ‘H’ (measured from top to bottom in the diagram) is the same as the width of the legs, and the cross bar is situated at the centre of the height of the ‘H’. This information can be seen in the diagram below.

q7-3-3-very-hard-trigonometry-ib-maths-

The ‘H’ is situated on horizontal ground but is tilted backwards slightly such that the rear face of the ‘H’ makes an angle of 5degree  to the vertical. During repair works, a metal support bar is connected between a point A on the ground and point M, which is the midpoint of the rear of the cross bar of the ‘H’. The plane defined by points A, M and B, where B is the midpoint of the gap between the legs of the ‘H’ where they touch the ground, is perpendicular to the plane defined by the rear face of the ‘H’. The angle straight A straight M with hat on top straight B  between the support bar and the rear face of the ‘H’ is 30°.

Calculate the length of the metal support bar.

7b
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8 marks

Additional support bars are required to reach from the ground to the midpoint at the top of the rear surface of each of the two legs. These additional supports are to contact the ground at the same point as the initial metal support bar. 

Calculate:

(i)
the length of one of the additional supports, and

(ii)
the angle that it makes with the horizontal.

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8a
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3 marks

A pitched roof is made up of a timber frame, ABCDEF, whose horizontal rectangular base ADFC covers an area of 15.3 m by 8.2 m. The raised central ridge BE is parallel to AD, and its midpoint is situated directly above the intersection of line segments AF and CD. BE is 12.1 m long and is at a height of 2.2 m above the plane defined by ADFC.

q8-3-3-very-hard-trigonometry-ib-maths-

Calculate the total length of timber required for the frame.

8b
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3 marks

An internal beam runs from the midpoint of AC, M, to point E.

Calculate the length ME.

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9a
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5 marks

In the diagram below, AB, BC and AC are steel beams at the first floor level of a building under construction. ABC lies in a horizontal plane 5 m above ground level, with AB = 7.9 m, BC = 5.8 m and AC = 8.3 m. The line of beam AB makes an angle of 78degree  with due north when viewed from above.

q9-3-3-very-hard-trigonometry-ib-maths-

A pot of paint has been left on beam BC, halfway between points B and C.

Find the bearing from point A to the pot of paint.

9b
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3 marks

A workman is standing on the second floor of the building directly above point A, with his eyes at a height of 12 m above ground level.

Find

(i)
the angle of depression, and
(ii)
the distance

from the workman’s eyes to the pot of paint.

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10
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6 marks

A tent with a symmetrical triangular cross-section, ABC, is fixed to a horizontal surface with guy ropes. The tent has a perpendicular height of 1.2 m, and the guy ropes are attached to points X and Y on AB and BC, respectively, such that BX = BY = 0.7 m. The guy ropes are each 1.1 m long, and they are anchored to the ground at points P and Q such that the angles straight P straight X with hat on top straight A and straight Q straight Y with hat on top straight C are both 30degree. Points A, B, C, P, Q, X and Y all lie in a single plane. A diagram to illustrate this is provided below. 

q10-3-3-very-hard-trigonometry-ib-maths-

Calculate the distance, PQ, between the anchor points of the two guy ropes.

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