GCSE Maths: Loci: the four basic rules (Higher)
Part of the GCSE Maths loci, bearings and vectors chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Loci: the four basic rules
A locus (the plural is loci) is the set of all the points that obey a rule. A locus is usually a line, a curve, or a region. For example, think of a dog tied to a post by a rope 4 m long. The dog can stand on any point that is 4 m or less from the post: the locus of the furthest points it can reach is a circle.
There are four basic loci. Every harder locus problem uses them.
1. A fixed distance from a point. The locus of points that are exactly 3 cm from a point A is a circle with centre A and radius 3 cm. The points inside the circle are less than 3 cm from A, and the points outside are more than 3 cm from A.
2. The same distance from two points A and B. This locus is the perpendicular bisector of AB (Section 22.2). The points on one side are nearer to A, and the points on the other side are nearer to B.
3. A fixed distance from a line. The locus of points exactly 1.5 cm from a straight line segment AB is made of two parallel lines (one each side of AB) with a semicircle at each end. It looks like a running track (a stadium shape). If the line goes on for ever, the locus is just two parallel lines, 1.5 cm from the line on each side.
4. The same distance from two lines. The points that are the same distance from two lines that cross lie on the bisector of the angle between the lines (Section 22.2). On one side of the bisector the points are nearer to the first line; on the other side they are nearer to the second line.
Reading the rules. Words such as “within 3 cm” or “no more than” mean the circle and everything inside it. Words such as “at least 3 cm from” mean the circle and everything outside. “Closer to A than to B” means the side of the perpendicular bisector that contains A. Use a solid line for a boundary that is included and a dashed line for one that is not.
Key ideas
- The locus of points a fixed distance from a point is a circle.
- The locus of points equidistant from two points is their perpendicular bisector.
- The locus of points a fixed distance from a line segment is a stadium shape (two parallel lines and two semicircles).
- The locus of points equidistant from two lines is the angle bisector.
- Within / less than = inside. At least / more than = outside.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The points exactly 3 cm from A make a circle. The points less than or equal to 3 cm are the inside of the circle.
- Mark the point A. This is the centre of the locus.Point A
- Set the compasses to 3 cm. Every point on the arc will be exactly 3 cm from A.Radius = 3 cm
- Draw the circle. Put the compass point on A and draw a complete circle.
- Shade the region. “Less than 3 cm” means the points inside the circle.
Check. A point 2 cm from A is inside the shaded region, and a point 4 cm from A is outside it. ✓
Watch out. The circle is the locus of points exactly 3 cm away. The inside of the circle is the locus of points less than 3 cm away.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The locus of points equidistant from A and B is the perpendicular bisector of AB. The points nearer to A are on the same side as A.
- Draw AB. Use a ruler. AB = 5 cm.AB = 5 cm
- Construct the perpendicular bisector. Use equal arcs from A and from B (more than 2.5 cm), then join the places where they cross (Section 22.2).Leave the arcs.
- Shade the correct side. The side that contains A is closer to A.
Check. Pick a point on the shaded side, such as A itself: it is 0 cm from A and 5 cm from B. ✓
Watch out. Do not shade the side with B on it.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The locus is two straight lines parallel to AB, joined at each end by a semicircle centred on the end of the segment.
- Draw AB. Use a ruler.AB = 4 cm
- Draw the two parallel lines. At A and at B, construct perpendiculars to AB, mark points 1.5 cm along each, and join them with a ruler (or use a set square).The lines are 1.5 cm above and 1.5 cm below AB.
- Draw the two semicircles. Put the compass point on A with radius 1.5 cm and draw a semicircle around the end. Repeat at B.Radius 1.5 cm at each end.
- Check the shape. It should look like a running track.
Check. Every point on the track is exactly 1.5 cm from the nearest point of AB. ✓
Watch out. Do not draw straight ends. Points near the end of AB must be 1.5 cm from B, which gives a semicircle.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The points equidistant from two lines are on the bisector of the angle between them.
- Draw the two lines. Draw the lines OA and OB with an angle of 60° at O.Angle AOB = 60°
- Construct the angle bisector. Draw an arc from O, then equal arcs from P and Q, then join O to where they cross (Section 22.2).
- Say what the line means. Every point on this line is the same distance from OA as from OB.The angle between each arm and the bisector is 30°.
Check. Take the point R on the bisector 4 cm from O. Its perpendicular distance to either arm is 4 sin 30° = 2 cm. ✓
Watch out. Measure the distance to a line along a perpendicular, not a slanting line.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. A mobile phone mast can be received up to 12 km away. The points within 12 km of the mast form a circle. On the map, divide by the scale.
- Change 12 km to cm. 1 km = 100 000 cm.12 × 100 000 = 1 200 000 cm
- Divide by the scale number. Real life to map: divide. The scale is 1 : 400 000.1 200 000 ÷ 400 000 = 3 cm
- Draw the locus. Draw a circle of radius 3 cm centred on the mast M. Shade the inside of the circle.
Check. A scale of 1 : 400 000 means 1 cm is 4 km, so 3 cm is 12 km. ✓
Watch out. The radius on the map is 3 cm, not 12 cm.
Watch out
A locus is the whole set of points, not one point. Students often draw only a single dot or a short line. Also decide carefully whether the region is the circle itself, the inside, or the outside: “at most” and “within” mean inside.Exercise 22E Basic loci
Non-calculatorKey words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The rule is “a fixed distance from a fixed point”.
- Use the rule. All the points the same distance from P form a circle with centre P.Radius = 5 cm
- Write the answer. A circle, centre P, radius 5 cm.
Check. Any point on the circle is 5 cm from P. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The rule is “equidistant from two points”.
- Use the rule. Points equally far from A and B lie on the line that cuts AB in half at 90°.Perpendicular bisector of AB
- Write the answer. The perpendicular bisector of AB.
Check. M, the midpoint of AB, is on the line and is the same distance from A and B. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The rule is “equidistant from two lines”.
- Use the rule. The line that cuts the angle between the lines into two equal parts.Angle bisector
- Write the answer. The bisector of the angle between the lines.
Check. Each point on it is the same perpendicular distance from both lines. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The rule is “a fixed distance from a line segment”.
- Use the rule. On each side of AB the locus is a line parallel to AB. Beyond each end it curves round.Parallel lines at 2 cm; semicircles of radius 2 cm at A and B
- Write the answer. A stadium (running track) shape.
Check. The ends are semicircles centred on A and B. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The tip stays the same distance from the centre.
- Find the fixed distance. The hand is rigid, so the tip is always the length of the hand from the centre.Fixed distance from the centre of the clock
- Write the answer. A circle centred on the middle of the clock face.
Check. After 60 seconds the tip is back where it started. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Equal distance from two points is a perpendicular bisector.
- Use the rule. The two lamp posts are the two fixed points.Same distance from A and B
- Write the answer. The perpendicular bisector of the line joining the two posts.
Check. She walks in a straight line, halfway between the posts. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The centre of the wheel stays the same height above the ground.
- Find the fixed distance. The centre is always one radius above the ground.30 cm
- Write the answer. A straight horizontal line 30 cm above the ground.
Check. The wheel can only roll along the ground, so its centre cannot go up or down. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The seat stays 2 m from the pivot.
- Find the fixed distance. The chain does not stretch.2 m from the bar
- Write the answer. An arc of a circle of radius 2 m, centred on the bar.
Check. The swing goes through part of a circle, so the locus is an arc. ✓
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Use compasses set to 3 cm, with the point on A.
- Mark A. It is the centre.Point A
- Set the compasses. The radius is the fixed distance.Radius = 3 cm
- Draw the circle and shade the inside. Less than 3 cm means inside the circle.
Check. A point inside is nearer than 3 cm. ✓
Watch out. Do not shade outside the circle.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The boundary has two straight parts and two semicircles. The two semicircles together make one whole circle of radius 2 cm.
- Find the straight parts. Each is the same length as AB.2 × 4 = 8 cm
- Find the curved parts. Two semicircles make one circle. Circumference = 2πr.2 × π × 2 = 12.57 cm
- Add. Straight parts plus curved parts.8 + 12.57 = 20.6 cm to 1 d.p.
Check. The track is longer than the 8 cm of straight sides. 20.6 cm is more than 8 cm. ✓
Watch out. Do not forget the curved ends.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Find the distance from the point to each of A and B using Pythagoras, then compare them.
- Work out the distance squared to A. Use Pythagoras: (horizontal difference)2 + (vertical difference)2. We compare squares, so there is no need for a square root.(3 − 1)2 + (5 − 1)2 = 4 + 16 = 20
- Work out the distance squared to B. Do the same for the second point.(3 − 7)2 + (5 − 1)2 = 16 + 16 = 32
- Compare. The point with the smaller distance squared is closer.20 < 32, so the point is closer to A.
Check. Distances: 4.47 and 5.66. The smaller one agrees. ✓
Watch out. Squaring does not change which distance is bigger, because distances are positive.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Find the distance from the point to each of A and B using Pythagoras, then compare them.
- Work out the distance squared to A. Use Pythagoras: (horizontal difference)2 + (vertical difference)2. We compare squares, so there is no need for a square root.(5 − 1)2 + (2 − 1)2 = 16 + 1 = 17
- Work out the distance squared to B. Do the same for the second point.(5 − 7)2 + (2 − 1)2 = 4 + 1 = 5
- Compare. The point with the smaller distance squared is closer.17 > 5, so the point is closer to B.
Check. Distances: 4.12 and 2.24. The smaller one agrees. ✓
Watch out. Squaring does not change which distance is bigger, because distances are positive.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Find the distance from the point to each of A and B using Pythagoras, then compare them.
- Work out the distance squared to A. Use Pythagoras: (horizontal difference)2 + (vertical difference)2. We compare squares, so there is no need for a square root.(4 − 1)2 + (7 − 1)2 = 9 + 36 = 45
- Work out the distance squared to B. Do the same for the second point.(4 − 7)2 + (7 − 1)2 = 9 + 36 = 45
- Compare. The point with the smaller distance squared is closer.45 = 45, so the point is equidistant from A and B.
Check. Distances: 6.71 and 6.71. The smaller one agrees. ✓
Watch out. Squaring does not change which distance is bigger, because distances are positive.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Find the distance from the point to each of A and B using Pythagoras, then compare them.
- Work out the distance squared to A. Use Pythagoras: (horizontal difference)2 + (vertical difference)2. We compare squares, so there is no need for a square root.(4.5 − 1)2 + (0 − 1)2 = 12.25 + 1 = 13.25
- Work out the distance squared to B. Do the same for the second point.(4.5 − 7)2 + (0 − 1)2 = 6.25 + 1 = 7.25
- Compare. The point with the smaller distance squared is closer.13.25 > 7.25, so the point is closer to B.
Check. Distances: 3.64 and 2.69. The smaller one agrees. ✓
Watch out. Squaring does not change which distance is bigger, because distances are positive.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. A point is within 3 units of C if its distance from C is at most 3.
- Work out the distance squared. Use Pythagoras between the point and C.(6 − 4)2 + (5 − 4)2 = 4 + 1 = 5
- Work out the radius squared. We compare with the radius squared.32 = 9
- Compare. The point is within 3 units of C if its distance squared is less than or equal to the radius squared.5 ≤ 9, so the point is inside (or on) the circle.
Check. Distance = √5 = 2.24, which is at most 3. ✓
Watch out. On the boundary (distance exactly equal to the radius) the point is still “within” if the rule says at most.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. A point is within 3 units of C if its distance from C is at most 3.
- Work out the distance squared. Use Pythagoras between the point and C.(1 − 4)2 + (1 − 4)2 = 9 + 9 = 18
- Work out the radius squared. We compare with the radius squared.32 = 9
- Compare. The point is within 3 units of C if its distance squared is less than or equal to the radius squared.18 > 9, so the point is outside the circle.
Check. Distance = √18 = 4.24, which is more than 3. ✓
Watch out. On the boundary (distance exactly equal to the radius) the point is still “within” if the rule says at most.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. A point is within 3 units of C if its distance from C is at most 3.
- Work out the distance squared. Use Pythagoras between the point and C.(7 − 4)2 + (4 − 4)2 = 9 + 0 = 9
- Work out the radius squared. We compare with the radius squared.32 = 9
- Compare. The point is within 3 units of C if its distance squared is less than or equal to the radius squared.9 ≤ 9, so the point is inside (or on) the circle.
Check. Distance = √9 = 3.00, which is at most 3. ✓
Watch out. On the boundary (distance exactly equal to the radius) the point is still “within” if the rule says at most.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. A point is within 3 units of C if its distance from C is at most 3.
- Work out the distance squared. Use Pythagoras between the point and C.(4 − 4)2 + (7.5 − 4)2 = 0 + 12.25 = 12.25
- Work out the radius squared. We compare with the radius squared.32 = 9
- Compare. The point is within 3 units of C if its distance squared is less than or equal to the radius squared.12.25 > 9, so the point is outside the circle.
Check. Distance = √12.25 = 3.50, which is more than 3. ✓
Watch out. On the boundary (distance exactly equal to the radius) the point is still “within” if the rule says at most.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. M is the midpoint of AB and the perpendicular bisector meets AB at 90°. So triangle AMP is right-angled, with hypotenuse AP.
- Find AM. M is half way along AB.8 ÷ 2 = 4 cm
- Spot the right-angled triangle. Angle AMP = 90°. AP is opposite the right angle, so it is the hypotenuse.AP = 5 cm, AM = 4 cm
- Use Pythagoras. Subtract the squares to find a shorter side.PM2 = 52 − 42 = 25 − 16 = 9
PM = 3 cm
Check. 3, 4, 5 is a Pythagorean triple. ✓
Watch out. Remember that M is the midpoint, so AM is 4 cm, not 8 cm.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The bisector makes 40° with each arm. Drop a perpendicular from P to OA to make a right-angled triangle, then use sin.
- Find the angle at O. The bisector cuts 80° into two equal parts.80° ÷ 2 = 40°
- Draw the right-angled triangle. The perpendicular from P meets OA at F. In triangle OFP, OP = 4 cm is the hypotenuse and PF is opposite the 40° angle.Opposite and hypotenuse means sin.
- Use sin. sin 40° = opposite ÷ hypotenuse, so opposite = hypotenuse × sin 40°.PF = 4 × sin 40° = 2.5712 = 2.57 cm
Check. The distance to the other arm is the same, 2.57 cm, because P is on the bisector. ✓
Watch out. Use the half-angle (40°), not 80°.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. The goat can reach every point within 6 m of the post. This is a circle of radius 6 m, so use area = πr2.
- Find the radius. The rope is as long as the radius.r = 6 m
- Use the formula. Area of a circle = π × radius × radius.π × 62 = π × 36 = 113.10
- Round. Nearest whole number.113 m2
Check. A square 12 m by 12 m has area 144 m2, and the circle (113) fits inside it. ✓
Watch out. The radius is the rope length, not the diameter.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the map.
- Change the real length into centimetres. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.25 km × 100 000 = 2 500 000 cm
- Say what the scale means. A scale of 1 : 500 000 means that every 500 000 cm in real life is only 1 cm on the map.500 000 cm real = 1 cm on the map
- Divide by 500 000. The map length is 500 000 times smaller than the real length.2 500 000 ÷ 500 000 = 5 cm
Check. Go forwards: 5 × 500 000 = 2 500 000 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Test a point that is not the midpoint. If it is also equidistant from A and B, then the locus is more than one point.
- Choose a point. Let A and B be 8 cm apart and let P be on the perpendicular line through the midpoint M, 3 cm above M.AM = MB = 4 cm and PM = 3 cm
- Find PA and PB. Both are hypotenuses of right-angled triangles with sides 4 and 3.PA = PB = √(42 + 32) = 5 cm
- Write the conclusion. P is the same distance from A and from B but is not the midpoint. So every point on the line through M at 90° to AB is in the locus.The locus is the whole perpendicular bisector.
Check. Choosing any other height above M gives another point of equal distance. ✓
Watch out. A locus is every point that obeys the rule.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. “At least 3 cm” means 3 cm or more.
- Test a point inside. A point at the centre is 0 cm from A.0 cm < 3 cm, so it does not obey “at least 3 cm”.
- Test a point outside. A point 5 cm from A.5 cm ≥ 3 cm, so it does obey the rule.
- Write the conclusion. The points that are 3 cm or more away are on and outside the circle.Tom should shade the outside of the circle (and the circle itself).
Check. The inside of the circle is the locus of points less than 3 cm from A. ✓
Watch out. At least means 3 or more. Within means 3 or less.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Half the distance from A to B is always half the radius. Find the distance AM.
- Say what stays fixed. B stays 4 cm from A.AB = 4 cm
- Find AM. M is the midpoint of AB, so it is half way from A to B.AM = 4 ÷ 2 = 2 cm
- Say what this means. M is always 2 cm from A, however B moves.The locus of M is a circle of radius 2 cm centred on A.
Check. When B is due east of A, M is 2 cm east of A. When B is north of A, M is 2 cm north of A. ✓
Watch out. Always ask: what is the fixed distance from what fixed point?
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. A point (x, y) is the same distance from A and B when its distance squared to A equals its distance squared to B.
- Write each distance squared. Use Pythagoras.To A: x2 + (y − 2)2
To B: (x − 6)2 + (y − 8)2 - Make them equal and expand. (y − 2)2 = y2 − 4y + 4, and so on.x2 + y2 − 4y + 4 = x2 − 12x + 36 + y2 − 16y + 64
- Cancel the squares. Subtract x2 and y2 from both sides.−4y + 4 = −12x − 16y + 100
- Collect the terms. Add 12x and 16y and subtract 4.12x + 12y = 96
- Simplify. Divide by 12.x + y = 8
Check. The midpoint of AB is (3, 5): 3 + 5 = 8. The gradient of AB is 1 and the line has gradient −1, so they are perpendicular. ✓
Watch out. The locus is a straight line: the perpendicular bisector of AB.