GCSE Maths: Combined loci and region problems (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)
Combined loci and region problems
Real problems often give more than one rule: “the treasure is within 5 m of the tree and nearer to the gate than to the wall”. Each rule gives a locus. The answer is the region where all the rules are true at once. You find it in five steps.
- Draw an accurate plan (use the scale you are given, and a ruler).
- Draw the first locus lightly: a circle, a bisector, a parallel line. Decide which side (or inside/outside) obeys the rule.
- Draw the second locus in the same way, and any others.
- Shade the overlap: only the part of the plan that obeys every rule.
- Check with a test point inside your region. Measure to see that it obeys each rule.
Think of each rule as cutting away the part of the plan that does not obey it. What is left is the answer. For a rule such as “within 4 m of A” you keep the inside of a circle. For “at least 2 m from the wall” you keep the part of the plan on the far side of a line 2 m from the wall. For “nearer to A than B” you keep one side of the perpendicular bisector. For “nearer to side AB than to side AD” you keep one side of the angle bisector.
If a region would be empty (no point obeys every rule) say so. For example, no point can be within 2 cm of A and at least 5 cm from A.
Key ideas
- Draw each locus separately, then shade the overlap of all of them.
- Within / at most = inside the circle or band. At least = outside.
- “Nearer to A than B”: the side of the perpendicular bisector with A on it. “Nearer to AB than AD”: one side of the angle bisector.
- Use a test point to check your shaded region.
- Use the scale: each square in the plans in this book is 1 unit (1 m or 1 km).
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: within 4 m of A
Rule 2: nearer to side AB than to side AD - Draw rule 1: within 4 m of A. Each square is 1 m, so set the compasses to 4 squares. Put the point on A and draw an arc (a quarter circle) inside the garden.
- Draw rule 2: nearer to side AB than to side AD. Points the same distance from AB and AD are on the bisector of angle DAB. Construct it (it is the diagonal 45° line from A). Points nearer to AB are below this line.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (2.4, 1) measured from the bottom left corner of the plan. Test each rule:
distance from A = 2.6 m, which is at most 4 m ✓
distance from AB = 1.0 m and from AD = 2.4 m, so it is closer to AB ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: nearer to the tree P than to the tree Q
Rule 2: at least 2 m from the tree P
Rule 3: within 3 m of the wall AB - Draw rule 1: nearer to the tree P than to the tree Q. Points the same distance from P and Q lie on the perpendicular bisector of PQ. Construct it (it is the vertical line half way between the trees). Shade the side with P.
- Draw rule 2: at least 2 m from the tree P. Draw a circle with centre P and radius 2 squares. “At least 2 m” means on the circle or outside it.
- Draw rule 3: within 3 m of the wall AB. Draw a line parallel to AB and 3 squares above it. “Within 3 m” means between this line and the wall AB.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (3, 1.2) measured from the bottom left corner of the plan. Test each rule:
distance from P = 2.3 m and from Q = 6.4 m, so it is closer to P ✓
distance from P = 2.3 m, which is at least 2 m ✓
distance from AB = 1.2 m, which is at most 3 m ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: at most 6 km from town X
Rule 2: at least 4 km from town Y
Rule 3: within 2 km of the straight road XY - Draw rule 1: at most 6 km from town X. Each square is 1 km. Draw a circle with centre X and radius 6 squares. “At most 6 km” means inside the circle.
- Draw rule 2: at least 4 km from town Y. Draw a circle with centre Y and radius 4 squares. “At least 4 km” means outside the circle.
- Draw rule 3: within 2 km of the straight road XY. The road runs along the line through X and Y. Draw lines 2 squares above and 2 squares below it. “Within 2 km” means between these lines.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (3.6, 4) measured from the bottom left corner of the plan. Test each rule:
distance from X = 0.6 m, which is at most 6 m ✓
distance from Y = 7.4 m, which is at least 4 m ✓
distance from the road XY = 0.0 m, which is at most 2 m ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: nearer to A than to B
Rule 2: within 5 m of C
Rule 3: nearer to side AB than to side AC - Draw rule 1: nearer to A than to B. Construct the perpendicular bisector of AB and shade the side that contains A.
- Draw rule 2: within 5 m of C. Draw a circle with centre C and radius 5 squares. Shade inside it.
- Draw rule 3: nearer to side AB than to side AC. Construct the bisector of angle BAC. Points nearer to AB are on the AB side of the bisector.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (3.3, 1.6) measured from the bottom left corner of the plan. Test each rule:
distance from A = 3.7 m and from B = 5.0 m, so it is closer to A ✓
distance from C = 4.4 m, which is at most 5 m ✓
distance from AB = 1.6 m and from AC = 2.2 m, so it is closer to AB ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Watch out
Do not shade each locus separately and stop. The answer is only where the shading overlaps. Also, check each inequality: “at least” means outside the circle, and “nearer to AB” means one particular side of the bisector, so test a point before you choose.Exercise 22F Region problems
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. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: within 4 m of A
Rule 2: within 2 m of side AB - Draw rule 1: within 4 m of A. Set the compasses to 4 squares and draw an arc centred on A.
- Draw rule 2: within 2 m of side AB. Draw a line parallel to AB and 2 squares above it. Shade between this line and AB.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (2, 0.9) measured from the bottom left corner of the plan. Test each rule:
distance from A = 2.2 m, which is at most 4 m ✓
distance from AB = 0.9 m, which is at most 2 m ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: within 5 m of C
Rule 2: nearer to C than to D - Draw rule 1: within 5 m of C. Set the compasses to 5 squares. Put the point on C and draw an arc inside the rectangle.
- Draw rule 2: nearer to C than to D. Construct the perpendicular bisector of CD. It is the vertical line half way between D and C. Shade the side with C.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (6.1, 2.8) measured from the bottom left corner of the plan. Test each rule:
distance from C = 2.9 m, which is at most 5 m ✓
distance from C = 2.9 m and from D = 6.5 m, so it is closer to C ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: within 4 m of the tree T
Rule 2: nearer to side AB than to side DC
Rule 3: at least 1 m from side AD - Draw rule 1: within 4 m of the tree T. Each square is 1 m. Draw a circle with centre T and radius 4 squares. Shade inside it.
- Draw rule 2: nearer to side AB than to side DC. AB and DC are parallel. The points the same distance from both are on a line half way between them. Shade the side nearer to AB.
- Draw rule 3: at least 1 m from side AD. Draw a line parallel to AD and 1 square to its right. “At least 1 m” means on or to the right of this line.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (4.9, 1.6) measured from the bottom left corner of the plan. Test each rule:
distance from T = 1.4 m, which is at most 4 m ✓
distance from AB = 1.6 m and from DC = 4.4 m, so it is nearer to AB ✓
distance from AD = 4.9 m, which is at least 1 m ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: within 6 m of Q
Rule 2: at least 2 m from side PS
Rule 3: nearer to side RS than to side RQ - Draw rule 1: within 6 m of Q. Set the compasses to 6 squares and draw an arc centred on Q.
- Draw rule 2: at least 2 m from side PS. Draw a line parallel to PS and 2 squares to its right. “At least 2 m” means on this line or to its right.
- Draw rule 3: nearer to side RS than to side RQ. Points the same distance from RS and RQ are on the bisector of angle SRQ. Construct it (a 45° line from R). Points nearer to RS are above this line.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (6.5, 3.5) measured from the bottom left corner of the plan. Test each rule:
distance from Q = 4.9 m, which is at most 6 m ✓
distance from PS = 6.5 m, which is at least 2 m ✓
distance from RS = 2.5 m and from RQ = 3.5 m, so it is closer to RS ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: within 5 km of the lighthouse L
Rule 2: nearer to the port P than to L
Rule 3: at least 2 km from the straight coastline AB - Draw rule 1: within 5 km of the lighthouse L. Each square is 1 km. Draw a circle with centre L and radius 5 squares. Shade inside it.
- Draw rule 2: nearer to the port P than to L. Construct the perpendicular bisector of LP and shade the side that contains P.
- Draw rule 3: at least 2 km from the straight coastline AB. Draw a line parallel to the coast and 2 squares above it. “At least 2 km” means on or above this line.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (7.9, 3.9) measured from the bottom left corner of the plan. Test each rule:
distance from L = 4.1 m, which is at most 5 m ✓
distance from P = 2.8 m and from L = 4.1 m, so it is closer to P ✓
distance from the coast AB = 3.9 m, which is at least 2 m ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. Draw each rule as a boundary line, then shade the region that obeys all the rules.
- Look at the plan and list the rules. Each rule gives a locus. The answer must obey every rule at the same time.Rule 1: within 7 m of B
Rule 2: nearer to C than to A
Rule 3: at least 1.5 m from side AB - Draw rule 1: within 7 m of B. Draw a circle with centre B and radius 7 squares. Shade inside it.
- Draw rule 2: nearer to C than to A. Construct the perpendicular bisector of AC and shade the side that contains C.
- Draw rule 3: at least 1.5 m from side AB. Draw a line parallel to AB and 1.5 squares above it. Shade above this line.
- Shade the region that obeys all the rules. Points that are in the right place for every rule are where the regions overlap.
Check. Pick a point in the shaded region, for example (5.8, 3) measured from the bottom left corner of the plan. Test each rule:
distance from B = 5.2 m, which is at most 7 m ✓
distance from C = 4.4 m and from A = 6.5 m, so it is closer to C ✓
distance from AB = 3.0 m, which is at least 1.5 m ✓
Watch out. A point must pass every test. Shade only the overlap, not all of each region separately.
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 the point against each rule in turn. It is in the region only if every rule is true.
- Test rule 1: within 4 m of A. The distance from A to the point must be at most 4 m.A is at (0, 0). The point is (3, 2).
Distance = √(32 + 22) = 3.61 m: at most 4 ✓ - Test rule 2: nearer to AB than to AD. AB is the line y = 0 and AD is the line x = 0. The distance to AB is the y-coordinate and the distance to AD is the x-coordinate.Distance to AB = 2 m, distance to AD = 3 m: nearer to AB ✓
- Decide. The point must obey both rules to be in the shaded region.Both rules are true, so the point is in the region.
Check. Compare with the shaded region in the diagram for the garden. ✓
Watch out. Check every rule, not just the first one.
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 the point against each rule in turn. It is in the region only if every rule is true.
- Test rule 1: within 4 m of A. The distance from A to the point must be at most 4 m.A is at (0, 0). The point is (1, 3).
Distance = √(12 + 32) = 3.16 m: at most 4 ✓ - Test rule 2: nearer to AB than to AD. AB is the line y = 0 and AD is the line x = 0. The distance to AB is the y-coordinate and the distance to AD is the x-coordinate.Distance to AB = 3 m, distance to AD = 1 m: nearer to AD, so the rule fails
- Decide. The point must obey both rules to be in the shaded region.At least one rule fails, so the point is not in the region.
Check. Compare with the shaded region in the diagram for the garden. ✓
Watch out. Check every rule, not just the first one.
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 the point against each rule in turn. It is in the region only if every rule is true.
- Test rule 1: within 4 m of A. The distance from A to the point must be at most 4 m.A is at (0, 0). The point is (3.5, 3).
Distance = √(3.52 + 32) = 4.61 m: more than 4, so the rule fails - Test rule 2: nearer to AB than to AD. AB is the line y = 0 and AD is the line x = 0. The distance to AB is the y-coordinate and the distance to AD is the x-coordinate.Distance to AB = 3 m, distance to AD = 3.5 m: nearer to AB ✓
- Decide. The point must obey both rules to be in the shaded region.At least one rule fails, so the point is not in the region.
Check. Compare with the shaded region in the diagram for the garden. ✓
Watch out. Check every rule, not just the first one.
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 the point against each rule in turn. It is in the region only if every rule is true.
- Test rule 1: within 4 m of A. The distance from A to the point must be at most 4 m.A is at (0, 0). The point is (4, 1).
Distance = √(42 + 12) = 4.12 m: more than 4, so the rule fails - Test rule 2: nearer to AB than to AD. AB is the line y = 0 and AD is the line x = 0. The distance to AB is the y-coordinate and the distance to AD is the x-coordinate.Distance to AB = 1 m, distance to AD = 4 m: nearer to AB ✓
- Decide. The point must obey both rules to be in the shaded region.At least one rule fails, so the point is not in the region.
Check. Compare with the shaded region in the diagram for the garden. ✓
Watch out. Check every rule, not just the first one.
Key words. A locus (plural loci) is the set of all the points that obey a rule. Equidistant means the same distance away.
Method. “Within 2 cm” means the distance from A is 2 cm or less. “At least 5 cm” means the distance is 5 cm or more.
- Write each rule as a distance. Call the distance from A to the point d cm.Rule 1: d ≤ 2. Rule 2: d ≥ 5.
- Look for a value that satisfies both. A number cannot be 2 or less and also 5 or more.No number d satisfies both.
- Write the conclusion. The two regions (inside the 2 cm circle and outside the 5 cm circle) do not overlap.Aisha is right: the region is empty.
Check. A point 1 cm from A obeys rule 1 but not rule 2. A point 6 cm from A obeys rule 2 but not rule 1. ✓
Watch out. When the overlap is empty, say “there is no such region”.
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 that is within 3 cm of both A and B is close to both. Use the shortest distance between A and B: a straight line.
- Use the triangle idea. For any point P, the path A to P to B is at least as long as the straight line AB. So PA + PB ≥ AB.PA + PB ≥ 10
- Add the two rules. If PA ≤ 3 and PB ≤ 3 then PA + PB is at most 6.PA + PB ≤ 3 + 3 = 6
- Compare. The sum cannot be both at least 10 and at most 6.6 < 10, so no point can obey both rules.
Check. The two circles of radius 3 cm centred on A and B do not touch, since 3 + 3 = 6 is less than 10. ✓
Watch out. Draw the two circles on the page. They are far apart, so they do not overlap.
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 region is a sector: a fraction of a whole circle. The 45° angle is 45 out of 360 of the whole turn.
- Find the fraction of the circle. The angle is 45°, out of 360° in a full circle.45 ÷ 360 = 18
- Find the area of the whole circle. Area = πr2 with r = 4.π × 42 = 50.265 m2
- Multiply. Take one eighth of it.50.265 ÷ 8 = 6.28 m2
Check. It must be less than the quarter circle (12.57 m2), and 6.28 is half of that. ✓
Watch out. The bisector of a 90° corner makes a 45° angle.
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 100 000.20 km × 100 000 = 2 000 000 cm
- Say what the scale means. A scale of 1 : 50 000 means that every 50 000 cm in real life is only 1 cm on the map.50 000 cm real = 1 cm on the map
- Divide by 50 000. The map length is 50 000 times smaller than the real length.2 000 000 ÷ 50 000 = 40 cm
Check. Go forwards: 40 × 50 000 = 2 000 000 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. Points nearer to B than to A are to the right of the perpendicular bisector x = 4. Within 5 units of A is the inside of the circle of radius 5. The region is the part of the circle cut off by the chord x = 4: a segment.
- Find the half-angle at the centre. The chord is at distance 4 from the centre A, and the radius is 5. In the right-angled triangle, cos(half-angle) = adjacent ÷ hypotenuse.cos θ = 4 ÷ 5 = 0.8
θ = cos−1(0.8) = 36.87° - Find the angle of the sector. The segment’s sector has angle 2θ.2 × 36.87 = 73.74°
- Find the area of the sector. Sector area = (angle ÷ 360) × πr2.(73.74 ÷ 360) × π × 25 = 16.088
- Find the area of the triangle. The triangle has two sides of 5 and the angle between them. Area = ½ ab sin C.½ × 5 × 5 × sin(73.74°) = 12.000
- Subtract. Segment = sector − triangle.16.088 − 12.000 = 4.09
Check. The segment fits inside the rectangle 2 units wide and 6 units tall (area 12), so about 4 is sensible. ✓
Watch out. Use the half-angle in the right-angled triangle, but the full angle for the sector.