# Solve Loci problems

In this worksheet, students will follow a set of instructions to enable them to draw the locus.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Ruler and Compass Constructions

Difficulty level:

### QUESTION 1 of 10

I took my first course on how to be a sketch artist this week.  I am already drawing large crowds.

Loci is drawing out something that satisfies certain conditions.

Locus is a path formed by a point which moves according to a certain rule. The plural is loci.

Lets have a look.

Example 1

Open your compass to 2 cm and draw around the point stated.  All points are the same distance (equidistant )from the centre.

Example 2

Measure 5 cm away from the wall and draw a horizontal line.  Each point on that line will be 5 cm away.

Example 3

To get the curved edges open your compass to 5 cm and draw an arc from each end of the line.

Example 4

A goat is tethered to a stake in the ground with a 4 m rope.

Draw a diagram to show the area that the goat can graze.

Use a scale of 1 cm = 1 m

A dog is on a chain 2 m long.

The chain is attached to a rail and the dog can move up and down.

The rail is 6 cm long.

Draw out the area the dog can roam.

Circle

Race track shape

Square

Rectangle

Jill's house is 3 km from school and 4 km from the motorway.  Where could Jill's house be?

Sketch this out using a scale of 1 cm = 1 km

Click on Confident or Unsure when you want to reveal the answer.

 Confident Unsure Jill's house

Two lighthouses A and B  both shed light for a distance of 20 km.

Show the area that both lighthouses shed light.

Use a scale of 1 cm = 5 km

What shapes will be drawn on your diagram?

Squares

Circles

Rectangles

Triangles

A dog is tethered by a 5 m long rope at the right hand corner of a rectangular shed.  Draw and shade the area that the dog can move.

Use a scale of 1 cm = 1 m

Choose an option to reveal the answer.

 Confident Unsure Sketch drawing

A goat is tethered to a tree on a rope that is 15 m long.  Another goat is tied to  a shed with a rope 12 m long.

Show the common area where both goats can roam.  Use a scale of 1 cm = 3 m.

A radio receiver has been erected but will only work if it is less than 8 km from its transmitter but more than 5 km away from some power lines.  Shade the region in which the radio receiver can be operated.

Use a scale of 1 cm = 2 km.

 Confident Unsure Loci Drawing

Bodgit and Scarper, a well known firm of builders are building a house for their client.

They have to abide by some planning regulations.

The house must be 10 m away from a road.

It must not be within 20 m of an electricity pylon.

It has to be within 60 m of the water supply.

A pipeline has to run equal distance from the electricity pylons.

Draw a diagram to show where the building can take place.

Use a scale of 1 cm = 10 m

Draw a line AB.

Draw the locus of the points that are 4 cm away from the line.

• Question 1

A goat is tethered to a stake in the ground with a 4 m rope.

Draw a diagram to show the area that the goat can graze.

Use a scale of 1 cm = 1 m

reveal
EDDIE SAYS
Lets not kid ourselves. Goats get into GCSE loci questions a lot. This question is quite straight forward as the goat is tethered into a stake in the ground so can walk around it. This should say to you that you are looking to draw a circle.
• Question 2

A dog is on a chain 2 m long.

The chain is attached to a rail and the dog can move up and down.

The rail is 6 cm long.

Draw out the area the dog can roam.

Race track shape
EDDIE SAYS
The rail should give you the clue that as the dog can move up and down a race track shape would be required. Just think race dogs have a track to run around.
• Question 3

Jill's house is 3 km from school and 4 km from the motorway.  Where could Jill's house be?

Sketch this out using a scale of 1 cm = 1 km

Click on Confident or Unsure when you want to reveal the answer.

 Confident Unsure Jill's house
EDDIE SAYS
This is a bit different as you have two things to draw out. There are two points that Jill's house can be, don't forget to mark the two.
• Question 4

Two lighthouses A and B  both shed light for a distance of 20 km.

Show the area that both lighthouses shed light.

Use a scale of 1 cm = 5 km

done
EDDIE SAYS
Is this question throwing a little light on the subject now? This is a pretty straightforward one. Apply the instructions to both lighthouses.
• Question 5

What shapes will be drawn on your diagram?

Circles
EDDIE SAYS
The fact that you are only given one measurement and three towns to apply this to should suggest circles.
• Question 6

A dog is tethered by a 5 m long rope at the right hand corner of a rectangular shed.  Draw and shade the area that the dog can move.

Use a scale of 1 cm = 1 m

Choose an option to reveal the answer.

 Confident Unsure Sketch drawing
EDDIE SAYS
The fact that the shed is rectangular should suggest that this is not a full circle.
• Question 7

A goat is tethered to a tree on a rope that is 15 m long.  Another goat is tied to  a shed with a rope 12 m long.

Show the common area where both goats can roam.  Use a scale of 1 cm = 3 m.

reveal
EDDIE SAYS
Were you able to draw the common ground for the goats to roam in? Be nice to the goats, they are just kids after all.
• Question 8

A radio receiver has been erected but will only work if it is less than 8 km from its transmitter but more than 5 km away from some power lines.  Shade the region in which the radio receiver can be operated.

Use a scale of 1 cm = 2 km.

 Confident Unsure Loci Drawing
EDDIE SAYS
Always have a go at loci questions. Just think carefully about what is being said. They usually involve circles and constructions of some description.
• Question 9

Bodgit and Scarper, a well known firm of builders are building a house for their client.

They have to abide by some planning regulations.

The house must be 10 m away from a road.

It must not be within 20 m of an electricity pylon.

It has to be within 60 m of the water supply.

A pipeline has to run equal distance from the electricity pylons.

Draw a diagram to show where the building can take place.

Use a scale of 1 cm = 10 m

reveal
EDDIE SAYS
How did you get on with this one? There is a lot of information to take in isn't there. remember not to rub out your construction lines as they show you have done things properly.
• Question 10

Draw a line AB.

Draw the locus of the points that are 4 cm away from the line.

reveal
EDDIE SAYS
Always use a compass and ruler for accuracy on this type of question. The locus needs to be equidistant from the line.
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