Bearings, Scale Drawing, Constructions & Loci (OCR GCSE Maths)

The diagram shows the position of a lighthouse  and a harbour .

The scale of the diagram is 1 cm represents 5 km.

Work out the real distance between  and .

Measure the bearing of from .

A boat  is 20km from  on a bearing of .

On the diagram, mark the position of boat with a cross (x).
Label it .

The diagram shows the positions of a lighthouse , a yacht  and a tanker  on a map.

Scale 1 cm represents 10 km

Measure the bearing of from .

The tanker, , sails 80km on a bearing of .

Find the distance, in km, between the tanker and the lighthouse when the tanker is closest to the lighthouse.

The diagram shows the position of two churches,  and .

Church  is on a bearing of  from church .
Church  is on a bearing of  from church .

In the space above, draw an accurate diagram to show the position of church .

Mark the position of church  with a cross ().
Label it .

Use ruler and compasses to bisect the angle at .
You must show all your construction lines.

Use ruler and compasses to construct the perpendicular from the point to the line
You must show all your construction lines.

Here is a map.
The map shows two towns, Burford and Hightown.

A company is going to build a warehouse.

The warehouse will be less than 30 km from Burford and less than 50 km from Hightown.

Shade the region on the map where the company can build the warehouse.

The diagram shows a garden in the shape of a rectangle.
The scale of the diagram is 1 cm represents 2m.

Scale: 1 cm represents 2 m

Irfan is going to plant a tree in the garden.
The tree must be

On the diagram, shade the region where Irfan can plant the tree.

Use ruler and compasses to construct the perpendicular from point  to the line .
  You must show all your construction lines.

Here is a scale drawing of an office.

   The scale is 1 cm to 2 metres.

A photocopier is going to be put in the office.
The photocopier has to be closer to  than it is to .
The photocopier also has to be less than 8 metres from .

Show, by shading, the region where the photocopier can be put.

The diagram shows the positions of three points,  and  on a map.

The bearing of  from  is .
Angle  is .

Work out the bearing of  from .

ABCD is a quadrilateral.

Construct the bisector of angle ABC.
Show all your construction lines.

Construct the perpendicular bisector of BC.

Show all your construction lines.

Shade the region which is

• nearer to BC than to AB

and

• nearer to B than to C.

Four points A, B, C and D are shown on the scale diagram below.

Scale: 1 cm represents 5 m

On the diagram, construct and mark the two points that are

• the same distance from A and B

and 

• 15 m from C.

Show all your construction lines.

The points A, B, C and D represent the four corners of Monty’s garden.
His garden is bounded by four straight fences A to B, B to C, C to D and D to A.

Monty wants to plant a tree in his garden at a place that satisfies the two conditions in part (a).

Explain why there is only one position where Monty can plant his tree.

The scale diagram below shows two cities, P and Q.

A plane departs from P at 09 47 and arrives at Q at 12 07.

Work out the average speed, in kilometres per hour, of the plane.

...................... km/h

Give one reason why your answer may be inaccurate.

Use ruler and compasses to construct the bisector of angle .
You must show all your construction lines.

The scale drawing shows the position of a hall and the position of a library.

Scale:  represents

A post box is  from the library on a bearing of 

Show the position of the post box on the scale drawing.
Mark the position with a cross  and label it 

Use your scale drawing to find

Using ruler and compasses, show the region inside the grid that is

less than 4 cm from A
and
nearer to B than to C.

Label the region R.
Show all your construction lines.

Here is a map.
The position of a ship, , is marked on the map.

Point  is on the coast.
Ships must not sail closer than 500 m to point .

The ship sails on a bearing of 

Will the ship sail closer than 500 m to point ?
You must explain your answer.

Here is a scale drawing of a rectangular garden .

Scale: 1 cm represents 1 metre.

Jane wants to plant a tree in the garden

at least 5m from point ,

nearer to  than to

and   less than 3m from .

On the diagram, shade the region where Jane can plant the tree.

Here is a map.
The map shows two airports, and .

Sophie is going to have a hotel built.

The hotel

will be closer to airport  than to airport .
will be less than 40km from airport .

On the map, shade the region where the hotel can be built.

The diagram shows the positions of three towns, Acton (), Barston () and Chorlton ().

Barston is 8 km from Acton on a bearing of .
Chorlton is 9km from Barston on a bearing of .

Find the bearing of Chorlton from Acton.
Give your answer correct to 1 decimal place.
You must show all your working.

The scale 1 cm represents 25 m can be written in the form 1 : k.
Find the value of k.

k = ........................................................

The scale drawing represents a harbour.

Scale: 1 cm represents 25 m

A boat leaves the harbour from point C and sails on a path that is equidistant from BC and CD.
The harbour rules do not allow boats to sail within 75 m of point E.

Find by construction whether the path of the boat will follow the harbour rules.
Show all your construction lines.

The scale diagram below shows towns, A, B and C.
Line AB represents the road from A to B and line AC represents the road from A to C.

A shopping centre is to be built so that it is

• nearer to the road from A to B than the road from A to C,
• less than 14 km from town C.

Using construction, shade the region where the shopping centre could be built.
Show all your construction lines.

Explain why the region found in part (a) may not be an appropriate site for the shopping centre.

The diagram shows the positions of two towns, Amton and Bisham.

The bearing of Bisham from Amton is b°.
The bearing of Amton from Bisham is 6b°.

Calculate the 3-figure bearing of Amton from Bisham.

A, B, C and D are four towns.

B is 25 kilometres due East of A.
C is 25 kilometres due North of A.
D is 45 kilometres due South of A.

Work out the bearing of B from C.

Calculate the bearing of D from B.

The scale drawing represents a park.

Scale: 1 cm represents 25m

A new play area must be

• no more than 150 m from B
• closer to AD than to CD.

Construct and shade the region where the play area can be positioned.

Show all your construction lines.

The diagram shows the positions of three ships, and .

Ship is due north of ship .

The bearing of ship from ship is 120°

Calculate the bearing of ship from ship .
Give your answer correct to the nearest degree.

,  and  are three towns.

The bearing of  from  is 105°
The bearing of from B is 230°

The distance of  from  is 180 km.
The distance of  from  is 95 km.

Calculate the distance of  from .
Give your answer correct to 3 significant figures.

........................km

and are ships.

is a port.

is due South of .
Angle

Work out the bearing of from .

A ship sails from to and then from to .

is 12 miles from , on a bearing of 080°

is 28 miles from , on a bearing of 155°

Work out the direct distance from to .

represents the plan of a field.

There is a path across the field that

starts at
is the same distance from and .

Using ruler and compasses, show the position of the path.

Use ruler and compasses to answer this question.

Point is

• the same distance from  and 
• 6 cm from .

Show the position of  on the diagram.