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Loci

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How to find all the points a set distance from a point or a line, or the same distance from two points or two lines, and how to shade a region on a scale plan that meets two conditions.

Loci in the exam

A locus is all the points that follow a rule, such as "exactly 3 cm from P". The plural of locus is loci. A locus can be a circle, a straight line, or a shape made of lines and arcs.

Loci questions usually come near the end of a calculator paper, for 3 or 4 marks. You are given a scale drawing of a garden, a room or a map, and asked to show, by shading, the region where something can go, or to show where a point could be.

There are four loci to know: - a fixed distance from a point - a fixed distance from a line - the same distance from two points - the same distance from two lines

Take a ruler and a pair of compasses into the exam. Draw each locus accurately, and leave your arcs on the drawing.

A fixed distance from a point

All the points exactly 3 cm from a point P make a circle with centre P and radius 3 cm. To draw it, open your compasses to 3 cm, put the point on P and draw all the way round.

Read the words carefully: - "less than 3 cm from P" or "up to 3 cm from P": inside the circle - "more than 3 cm from P" or "at least 3 cm from P": outside the circle

Points on the circle itself count for "up to" and "at least".

On a scale drawing, change the real distance to a distance on the drawing first. With a scale of 1 cm to 5 m, a real distance of 15 m is 15\div5=3 cm on the drawing.

Worked example 1: A circle on a scale drawing

A goat is tied to a post P with a rope 8 m long. Ali makes a scale drawing, using 1 cm to represent 2 m. Describe the region on his drawing that the goat can reach.

  1. The goat can reach every point up to 8 m from P, in any direction.
  2. The points exactly 8 m from P make a circle with centre P.

  3. Your turn. On the drawing, 1 cm represents 2 m. Work out 8\div2

    Radius on the drawing =8\div2=4 cm

  4. Open your compasses to 4 cm, put the point on P and draw the circle. "Up to 8 m" means on or inside the circle, so shade the inside.
  5. Answer: the inside of a circle with centre P and radius 4 cm

A fixed distance from a line

Distance from a line is measured at right angles to the line. That is the shortest way from a point to the line.

All the points exactly 2 cm from a long straight line make two lines, parallel to it, one on each side, each 2 cm away.

If the line has ends, like a line AB, the locus also goes round each end in a semicircle of radius 2 cm, centred on A and on B. The whole shape looks like a running track.

On a plan of a room or a garden, a wall or a fence is a line. Only the inside of the plan matters, so you draw just one line, parallel to the wall.

Worked example 2: A distance from a hedge

The plan of a rectangular garden ABCD is drawn to a scale of 1:50. A hedge runs along the side AB. A vegetable patch must be more than 1.5 m from the hedge. Show on the plan where the vegetable patch can go.

  1. The points exactly 1.5 m from the hedge make a line parallel to AB, inside the garden.
  2. Change to centimetres: 1.5\times100=150 cm

  3. Your turn. The scale is 1:50, so divide by 50. Work out 150\div50

    On the plan, the line is 150\div50=3 cm from AB

  4. Draw a line parallel to AB, 3 cm from it. "More than" means further from the hedge, so shade the side away from AB.
  5. Answer: the part of the garden more than 3 cm from AB on the plan, as shaded

The same distance from two points

Equidistant means the same distance. All the points equidistant from A and B lie on the perpendicular bisector of AB: the line that cuts AB in half, at right angles.

Points on A's side of this line are closer to A than to B. Points on B's side are closer to B.

To draw it, open your compasses to more than half of AB. Draw an arc above and below AB with the point on A. Without changing the compasses, do the same from B. Join the two points where the arcs cross.

On a grid, when AB is across the page, the perpendicular bisector is the straight line up the page through the midpoint of AB.

Worked example 3: Equidistant from two points on a grid

A is the point (1, 2) and B is the point (7, 2). A lamp post L is on the line y=5. L is the same distance from A as it is from B. Work out the coordinates of L.

  1. Points the same distance from A and B lie on the perpendicular bisector of AB.
  2. AB goes across the grid, so its perpendicular bisector goes straight up through the midpoint of AB.

  3. Your turn. What number is halfway between 1 and 7?

    The midpoint of AB is (4, 2), so the perpendicular bisector is the line x=4

  4. L is where the line x=4 meets the line y=5
  5. Check: from A to L is 3 across and 3 up. From B to L is also 3 across and 3 up.
  6. Answer: L is at (4, 5)

The same distance from two lines

All the points equidistant from two lines that meet lie on the angle bisector: the line that cuts the angle between them exactly in half.

Points on one side of the angle bisector are closer to the line on that side.

To draw it, put the compass point on the corner C and draw an arc that crosses both lines. From each of those two crossings, draw an arc inside the angle, keeping the compasses the same. Join C to the point where these two arcs cross.

Worked example 4: Equidistant from two fences

Two straight fences meet at a corner C. The angle between them is 84^\circ. A row of lamps is put in so that each lamp is the same distance from both fences. (a) What line are the lamps on? (b) Work out the angle between the row of lamps and each fence.

  1. Points the same distance from two lines that meet are on the bisector of the angle between them.
  2. (a) The lamps are on the bisector of the angle at C.
  3. (b) The angle bisector cuts the 84^\circ angle into two equal parts.

  4. Your turn. Work out 84\div2

    84\div2=42^\circ

  5. Answer: (a) the bisector of the angle at C (b) 42^\circ

Regions with two conditions

Many questions have two conditions, such as "closer to A than to B, and less than 4 cm from C". Draw the locus for each condition, decide which side of it you need, then shade only the part where both are true.

  • Closer to A than to B: A's side of the perpendicular bisector of AB
  • Less than 4 cm from C: inside the circle with centre C and radius 4 cm
  • More than 2 m from a wall: the far side of a line parallel to the wall

The edges of your shaded region should be the loci you drew, and the sides of the plan. Examiners see many students shade an area with no loci drawn at all, which gets no marks for the region.

To test whether a point is in a region, check it against each condition in turn. It must pass every one.

Worked example 5: Shading a region in a garden

The plan of a rectangular garden ABCD is drawn to a scale of 1:200. On the plan, AB is 8 cm and AD is 6 cm. A bird bath is to be put in the garden. It must be closer to A than to B, and less than 10 m from D. Show, by shading, the region where the bird bath can go.

  1. Closer to A than to B: draw the perpendicular bisector of AB. AB is 8 cm, so the bisector is 4 cm from A, at right angles to AB. The bird bath is on A's side.
  2. Less than 10 m from D: change to centimetres, 10 m =1000 cm

  3. Your turn. The scale is 1:200, so divide by 200. Work out 1000\div200

    Radius on the plan =1000\div200=5 cm

  4. Draw an arc with centre D and radius 5 cm. Shade the part that is inside the arc and on A's side of the bisector.
  5. Check a point: the middle of AD is 3 cm from D, less than 5 cm, and it is on A's side. It is in the shaded region.
  6. Answer: the region inside the arc with centre D and radius 5 cm, on A's side of the perpendicular bisector of AB

Worked example 6: A point that meets two conditions

A plan of a park is drawn on a centimetre grid, using 1 cm to represent 10 m. Two straight paths meet at right angles at the corner O. On the plan they lie along the x-axis and the y-axis. Bench P is at (2, 9) and bench Q is at (8, 9). A statue S is to be put the same distance from both paths, and the same distance from P as from Q. (a) Work out the coordinates of S. (b) How far is the statue from each path in real life?

  1. (a) The same distance from both paths: S is on the bisector of the right angle at O. This line goes through (1, 1), (2, 2), (3, 3) and so on. It is the line y=x
  2. The same distance from P as from Q: S is on the perpendicular bisector of PQ. PQ goes across the grid, so this line goes straight up through the midpoint of PQ, (5, 9). It is the line x=5

  3. Your turn. S is where the line x=5 meets the line y=x. Write down the coordinates of S.

    Write your answer as (x, y)

    S is at (5, 5)

  4. Check: S is 5 across from one path and 5 up from the other. From P to S is 3 across and 4 down, and from Q to S is also 3 across and 4 down.

  5. Your turn. (b) On the plan, S is 5 cm from each path, and 1 cm represents 10 m. How far is S from each path in real life, in metres?

    5\times10=50 m

  6. Answer: (a) (5, 5) (b) 50 m

Common mistakes

  • Marking a few points instead of drawing the locus. Draw the whole circle or arc with compasses, so every point on it is the right distance.
  • Forgetting the scale. Change the real distance to a distance on the drawing before you set your compasses.
  • Drawing a square round a point. The points a fixed distance from a point make a circle, not a square.
  • Shading the wrong side. "Less than" means inside the circle, and "more than" or "at least" means outside it.
  • Guessing the right angle of a bisector. Use compasses and leave your arcs, so the line is accurate. If the question says "Use ruler and compasses", the arcs must be shown.
  • Shading a region with no edges. Draw every locus first, then shade only the part where all the conditions are true.

Practice questions

More on this topic: Loci worksheet with full solutions