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Solve Loci problems

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

'Solve Loci problems' worksheet

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:  

Worksheet Overview

QUESTION 1 of 10

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

sketch pad

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.

 

eye

Lets have a look.

Example 1

Locus

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

locus example

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

 

Example 4

Locus example

Question Ready to have a go?

goat

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

Type reveal into the answer box when you are ready to check your answer.

 

dog

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.

What shape is your sketch?

 

 

Circle

Race track shape

Square

Rectangle

red house

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.

 

 ConfidentUnsure
Jill's house

lighthouse

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

Type 'done' in the answer box when ready to see the answer

 

loci questoin

What shapes will be drawn on your diagram?

Squares

Circles

Rectangles

Triangles

dog

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.

 

 

 ConfidentUnsure
Sketch drawing

goats

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.

Type reveal in the answer box when you are ready to see the answer.

 

electricity pylons

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.

 

 

 ConfidentUnsure
Loci Drawing

 

builder

 

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

Type 'reveal' in the answer box when you are ready to view the answer.

 

pen

Draw a line AB.

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

Type 'reveal' in the answer box when you are ready to view the answer.

 

 

  • Question 1

goat

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

Type reveal into the answer box when you are ready to check your answer.

 

CORRECT ANSWER
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

dog

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.

What shape is your sketch?

 

 

CORRECT ANSWER
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

red house

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.

 

CORRECT ANSWER
 ConfidentUnsure
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

lighthouse

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

Type 'done' in the answer box when ready to see the answer

 

CORRECT ANSWER
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

loci questoin

What shapes will be drawn on your diagram?

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

dog

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.

 

 

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

goats

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.

Type reveal in the answer box when you are ready to see the answer.

 

CORRECT ANSWER
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

electricity pylons

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.

 

 

CORRECT ANSWER
 ConfidentUnsure
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

 

builder

 

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

Type 'reveal' in the answer box when you are ready to view the answer.

 

CORRECT ANSWER
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

pen

Draw a line AB.

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

Type 'reveal' in the answer box when you are ready to view the answer.

 

 

CORRECT ANSWER
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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