Constructions and loci

Construct the perpendicular from the point P to the line AB. Show all of your construction lines.

Construct the perpendicular from the point P to the line AB. Show all of your construction lines.

The diagram shows a rectangular garden ABCD.

A B

D C

The region of the garden that is closer to DA than DC is to be made into a patio.

Construct and shade the region for the patio.

Show all your construction lines.

The diagram represents a rectangular field. A and B are two trees.

A straight path goes across the field. The path is always the same distance from A and B.

Construct the route followed by the path. Show all your construction lines.

Using ruler and compasses only, construct the perpendicular bisector of the line AB.

Leave your construction lines visible.

Triangle ABC is drawn below.

Using a ruler and compasses only, construct and shade the region which is closer to AB than

AC.

Using ruler and compasses only, construct the road from town T to the road AB.

The new road costs £200 000 per kilometre to build. The road constructor says

The new road will cost over £600 000 to build.

Show that the road constructor is incorrect.

Using ruler and compasses only, construct the road from town T to the road AB.

The new road costs £200 000 per kilometre to build. The road constructor says

The new road will cost over £600 000 to build.

Show that the road constructor is incorrect.

The diagram represents a coastline.

A, B and C are lighthouses.

A boat is

• the same distance from A and B

• the same distance from AB and BC.

Using a ruler and compasses only, construct the position of the boat.

Label the position of the boat clearly.

Construct the perpendicular from the point P to the line AB.

The diagram shows a field LMN.

A tree is to be planted in the field so that it is

• the same distance from the fences MN and ML

and

• the same distance from corner M as from corner N.

Show, by construction, whether this can be done or cannot be done.

Construct the perpendicular from the point P to the line AB.

The diagram shows a field LMN.

A tree is to be planted in the field so that it is

• the same distance from the fences MN and ML

and

• the same distance from corner M as from corner N.

Show, by construction, whether this can be done or cannot be done.

The diagram shows the scale drawing of a sandpit, ABCD. It also shows the arc of all points in the sandpit that are 80 cm from corner A.

Scale: 1 cm represents 20 cm

A game is played by throwing a ball into the sandpit. Points may be scored when the ball lands in the sandpit.

• 1 point if the ball lands within 80 cm of corner A, and

• 1 point if the ball is closer to side AB than side AD, and

• 1 point if the ball is closer to corner A than corner B.

By completing the construction, find and shade the regions where 2 points can be scored.

Show all your construction lines.

The diagram shows the scale drawing of a sandpit, ABCD. It also shows the arc of all points in the sandpit that are 80 cm from corner A.

Scale: 1 cm represents 20 cm

A game is played by throwing a ball into the sandpit. Points may be scored when the ball lands in the sandpit.

• 1 point if the ball lands within 80 cm of corner A, and

• 1 point if the ball is closer to side AB than side AD, and

• 1 point if the ball is closer to corner A than corner B.

By completing the construction, find and shade the regions where 2 points can be scored.

Show all your construction lines.

The scale drawing represents a park, ABCD.

Scale: 1 cm represents 10 m

A straight path goes across the park from B. The path is always the same distance from side AB and side BC.

Construct the route followed by the path.

Show all your construction lines.

A bench is to be placed on the path. The bench must be no more than 50 m from C.

Construct the locus of the possible positions of the bench.

Indicate clearly on the diagram where the bench can be placed.

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.

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.

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

Find the value of k.

The scale drawing represents a harbour.

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.

Construct the perpendicular from the point P to the line AB. Show all of your construction lines.

Construct the perpendicular from the point P to the line AB. Show all of your construction lines.

The scale drawing shows Katy’s garden ABCD.

Scale: 1 cm represents 5 m

A B

D C

Katy places a statue in the garden.

The statue is

• more than 30 m from D

• closer to CB than AB.

Construct and shade the region where the statue could be placed.

Show all your construction lines.

Using ruler and compasses only, construct the perpendicular bisector of FG.

The diagram shows triangle ABC.

Construct the bisector of angle BAC.

Construct the perpendicular bisector of AC.

Shade the region inside triangle ABC that is

• nearer to AC than to AB

• nearer to A than to C.

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 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.

Scale: 1 cm represents 2 km

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

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

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

The diagram shows a scale drawing of a park, ABCD.

Scale: 1 cm represents 10 m

A straight water pipe runs across the park. The pipe runs equidistant from DA and DC.

Construct, using compasses and ruler only, the position of the water pipe.

You must show all your construction lines.

A straight path connects the entrance to the exit. This path is perpendicular to CB.

Construct, using compasses and ruler only, the position of the path.

Leave in all your construction lines.

Find the actual length of the path, in metres.

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

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

The scale diagram shows the positions of town A and town B.

Scale: 1 cm represents 10 miles

Lucy’s house is nearer to town A than to town B. Her house is exactly 50 miles from town B.

On the scale diagram show all the possible positions of Lucy’s house.

You must show all your construction lines.

The scale of a map is 1 cm represents 25 m.

The length of a path is 240 m.

Work out the length, in centimetres, of the path on the map.

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

Find the value of k.

The scale drawing represents a park.

Scale: 1 cm represents 25 m

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 scale of a map is 1 cm represents 25 m.

The length of a path is 240 m.

Work out the length, in centimetres, of the path on the map.

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

Find the value of k.

The scale drawing represents a park.

Scale: 1 cm represents 25 m

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.