Loci: combining rules and regions
The Maths Habit
Which construction?
The same distance from two lamp posts
Perpendicular bisectorExactly 3 km from a beacon
A circleThe same distance from two fences that meet at a corner
Angle bisector
A goat is tied to the corner C of a shed with a 5 m rope. Sam says it can graze the whole circle of radius 5 m. Is Sam right?
No. The rope bends round the shed: three quarters of a circle and two small arcs.
- Region
- The part of a drawing where every point fits a rule.
- Scale
- How a length on the plan compares with the real length.
- Locus
- All the points that follow a rule.
Two or more rules
- Change each distance with the scale.
- Draw each locus.Rule, locus, on the plan: fill in the table.
- Shade the region that fits every rule.Less than: inside. Nearer to: that side.
What radius do you draw?
Inside or outside?
A mobile mast M reaches 25 km (1 cm stands for 10 km). Shade where a phone gets the signal and is closer to A than to B. Inside the circle, on A's side
| Rule | Locus | On the plan |
|---|---|---|
| up to 25 km from M | circle | radius 2.5 cm |
| closer to A than B | perpendicular bisector | A's side |
Treasure is less than 20 m from the tree T and nearer to the wall AB than to the wall AC (1 cm stands for 10 m). Inside the circle, on AB's side
| Rule | Locus | On the plan |
|---|---|---|
| less than 20 m from T | circle | radius 2 cm |
| nearer to AB than AC | angle bisector | AB's side |
A lifeguard at L sees up to 25 m. Swimmers stay less than 10 m from the beach AB (1 cm stands for 10 m). Shade where they can swim. The strip inside the circle
| Rule | Locus | On the plan |
|---|---|---|
| up to 25 m from L | circle | radius 2.5 cm |
| less than 10 m from AB | parallel line | 1 cm from AB |
A walker gets a phone signal up to 30 km from mast P and up to 20 km from mast Q (1 cm stands for 10 km). Shade where the walker gets both signals. Inside both circles
| Rule | Locus | On the plan |
|---|---|---|
| up to 30 km from P | circle | radius 3 cm |
| up to 20 km from Q | circle | radius 2 cm |
A tent goes closer to the tap T than to the bin B, and more than 15 m from the hedge PQ (1 cm stands for 10 m). Shade where it can go. On T's side, beyond the parallel line
| Rule | Locus | On the plan |
|---|---|---|
| closer to T than B | perpendicular bisector | T's side |
| more than 15 m from PQ | parallel line | 1.5 cm from PQ |
A bird table goes nearer to the fence AB than to the fence AC, and more than 20 m from the gate K (1 cm stands for 10 m). Shade where it can go. On AB's side, outside the circle
| Rule | Locus | On the plan |
|---|---|---|
| nearer to AB than AC | angle bisector | AB's side |
| more than 20 m from K | circle | radius 2 cm, outside |
Back to the goat. The rope is 5 m and the shed is 4 m by 2 m. How much rope is left at each far corner?
5 − 4 = 1 m round one corner, 5 − 2 = 3 m round the other.
A goat is tied to the corner C of a shed with a 5 m rope. The shed is 4 m by 2 m (1 cm stands for 2 m). Shade where it can graze.
A beacon is the same distance from A as from B, and exactly 20 km from C (1 cm stands for 10 km). Mark the places it could be. Two points
Answer sheet
Thumbs up if it is right. Thumbs down if it is wrong.
Less than 3 cm from A: shade inside the circle.
RightWith two rules, shade where either rule works.
Wrong25 km at 1 cm to 10 km is a radius of 25 cm.
WrongA goat tied to a shed corner grazes three quarters of a circle and more round the corners.
Right
Kim shades the points less than 2.5 cm from P and closer to Q than to P. She shades the side with P. What is wrong?
Q's sideA shop is closer to A than B, less than 11 km from C and more than 4 km from the road. How many loci do you draw?
ThreeWhere are the points equidistant from A and B and 2 cm from AB, when AB = 6 cm?
Two points
A mast reaches 30 km, 1 cm stands for 10 km. Construction and radius?
A circle, radius 3 cmGoat: 5 m rope, shed 4 m by 2 m. Rope left round each far corner?
1 m and 3 mLess than 3 squares from A and nearer to B: shade.
Inside the circle, on B's side