GCSE · Constructions & loci

Loci & regionsExam-style questions

A locus as the set of points obeying a rule, the four standard loci, and how to combine several conditions — with scale drawings — to shade the region that satisfies all of them at once.

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11 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A goat is tied to a post $P$ in the middle of a large field by a rope of length $9$ m.

A plan of the field is drawn using a scale of $1$ cm to $2$ m.

(a) Work out the radius, in centimetres, of the circle that should be drawn on the plan to show how far the goat can reach.

(b) Describe the region of the plan that represents the grass the goat can eat.
Level 2 · 3 marksTry it
2
A plan is drawn to a scale of $1$ cm to $5$ m.

(a) A water sprinkler reaches every point within $20$ m of the tap $T$ . Work out the radius of the circle that shows this region on the plan.

(b) Describe fully the locus of points that are exactly $20$ m from $T$ (in the real garden).
Level 2 · 4 marksTry it
3
The diagram shows a plan of a rectangular shed measuring $30$ m by $15$ m, drawn to a scale of $1$ cm to $5$ m.A security light lights up all the ground within $20$ m of the outside walls of the shed.

On the plan, draw accurately the locus of the points that are exactly $20$ m from the outside of the shed.
Level 3 · 4 marksTry it
4
A treasure map is drawn to a scale of $1$ cm to $4$ m.

On the map, the treasure is buried at the point that is equidistant from a rock $R$ and a well $W$ , and also equidistant from two straight paths that meet at a point $T$ (measured within the acute angle between the paths).

(a) Name the construction needed for each of the two conditions.

(b) A student measures the distance from $R$ to the treasure on the map as $5.5$ cm. Work out the real distance from the rock to the treasure.
Level 3 · 4 marksTry it
5
Two statues, $A$ and $B$ , stand $12$ m apart in a park. A fountain is to be built so that it is exactly the same distance from $A$ as from $B$ .

(a) Describe fully the locus of possible positions for the fountain.

A plan of the park is drawn to a scale of $1$ cm to $2$ m.

(b) Work out the length of $AB$ on the plan.
Level 3 · 3 marksTry it

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