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
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
2
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
3
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
4
$ABCD$ is a rectangular room, with the corner $A$ where walls $AB$ and $AD$ meet.

A sensor must be placed so that it is closer to wall $AB$ than to wall $AD$ .

(a) Describe the boundary of the region where the sensor can go, and how you would construct it.

The room measures $8$ m by $6$ m. A plan is drawn to a scale of $1$ cm to $2$ m.

(b) Work out the dimensions of the room on the plan.
Level 3 · 4 marksTry it
5
The line segment $AB$ is drawn on a centimetre grid. $T$ is a point that is $2$ cm or less from the line segment $AB$ .

Zara says that $T$ must be inside the shaded rectangle.

Explain the mistake Zara has made.

(1 mark)
Level 3 · 1 markTry it