Loci

Think about what you would draw with a pair of compasses opened to 3 cm.

Count squares from P. Exactly 3 cm is not less than 3 cm.

"At least 2 cm" means 2 cm or more.

The line you want is the perpendicular bisector of AB.

The points the same distance from A and B lie on a line at right angles to AB, through the point halfway between them.

Points on the dashed line are the same distance from A and B. Which side of it is nearer to A?

1. Every point on the dashed line is the same distance from A as from B, so Q is not closer to A.
2. Points on the same side of the line as A are closer to A. Points on B's side are closer to B.
3. R and S are on B's side. P is on A's side, so P is closer to A than to B.

Think about walking alongside the road, always staying 20 m away from it. Could you do that on both sides?

1. A point 20 m from the road can be on either side of it.
2. On each side, the points 20 m away make a straight line running alongside the road, parallel to it.
3. So the points make two parallel lines, one on each side of the road, each 20 m from it.

The points the same distance from two lines that meet lie on the line that cuts the angle between them exactly in half.

1. Points the same distance from two crossing lines lie on the bisector of the angle between them.
2. The angle bisector starts at the junction and splits the angle between the roads into two equal angles.
3. Line M is nearer to Road 2 than to Road 1, and line N does not go through the junction. Line L splits the angle in half, so the trees go along line L.

How many lots of 5 km are there in 30 km?

1. Each centimetre on the map represents 5 km.
2. 30\div5=6
3. The circle should have a radius of 6 cm.

Count the squares from P out to the circle.

1. The points exactly 2 cm from P make a circle with centre P and radius 2 cm.
2. Ellie's circle does have centre P, but it reaches 4 squares from P, so its radius is 4 cm.
3. Her mistake is that the radius is 4 cm instead of 2 cm.

Check both conditions for each point: is it inside the circle, and is it on B's side of the dashed line?

1. Less than 3 cm from A means inside the circle. That rules out R and S.
2. Closer to B than to C means on B's side of the dashed line. That rules out Q.
3. P is inside the circle and on B's side of the line, so the lamp could be at P.

Change 5 m to centimetres first. A scale of 1 : 200 means 1 cm on the plan is 200 cm in the garden.

Part (a)

1. 5\text{ m}=500\text{ cm}
2. Each centimetre on the plan is 200 cm in the garden: 500\div200=2.5
3. The radius on the plan is 2.5 cm.

Part (b)

1. In the garden the bench is 2.2\times200=440 cm from the sprinkler.
2. 440\text{ cm}=4.4\text{ m}, which is less than 5 m.
3. Yes, the sprinkler waters the bench.

First find the line of points that are the same distance from A and B. Where does it cross the circle?

1. A is at (2,8) and B is at (6,8). The points the same distance from A and B lie on the perpendicular bisector of AB, the vertical line x=4.
2. The line x=4 goes through C(4,3), so it crosses the circle 3 units above and 3 units below C: at (4,6) and (4,0).
3. (4,0) is on the x-axis, not above it. T is (4,6).

Less than 30 m from B means inside B's circle. More than 30 m from A means outside A's circle.

1. Less than 30 m from tap B means inside the circle with centre B: regions Q and R.
2. More than 30 m from tap A means outside the circle with centre A: regions R and S.
3. The only region in both lists is R.

Count the radius of Ravi's circle in squares. How many centimetres on the drawing should 6 m be?

1. "At least 6 m" means 6 m or more, so the goat is allowed on or outside a circle round F. Shading outside is right.
2. 1 cm represents 2 m, so 6 m is 6\div2=3 cm on the drawing.
3. Ravi's circle has a radius of 6 squares, which is 6 cm. It should have a radius of 3 cm.

Region R is inside the circle and above the dashed line. Find each point on the grid and check both conditions.

1. (3,4) is on the line y=4, not above it.
2. (3,6) is on the circle, so it is exactly 3 units from A. That is not less than 3 units.
3. (6,5) is above the line but outside the circle, so it is more than 3 units from A.
4. (4,5) is above the line and inside the circle. So (4,5) is in region R.

Start with the line of points that are the same distance from A and B. Which points on it are 2 units from the fence?

1. A is (2,1) and B is (6,1). The points the same distance from A and B lie on the perpendicular bisector of AB, the vertical line x=4.
2. The points 2 units from the fence lie on two lines parallel to it, y=2 and y=6. On the line x=4 these give (4,2) and (4,6).
3. C is at (4,7). (4,2) is 5 units from C, but (4,6) is only 1 unit from C.
4. T is (4,6).

Count squares from M, then use the scale: each centimetre on the map is 15 km.

Part (a)

1. The edge of the shaded region is 4 squares from M, so the radius is 4 cm on the map.
2. 4\times15=60
3. The signal reaches up to 60 km from M.

Part (b)

1. T is 6 squares from M, so it is 6 cm from M on the map.
2. 6\times15=90, so T is 90 km from M.
3. Before, the signal reached 60 km. 90-60=30, so it now reaches 30 km further.

Each square is 1 m. Count the squares from B to each flower bed, along the grid lines.

1. Count from B along the grid lines. Each square is 1 m.
2. S is 1 m from B, R is 2 m from B and Q is 3 m from B. These are all less than 4 m, so they are still watered.
3. P is 4 m from B. Exactly 4 m is not less than 4 m, so P is no longer watered.
4. The answer is P.

Change both distances to centimetres on the map. Then check each point against both conditions.

Part (a)

1. Each centimetre represents 20 m.
2. 60\div20=3
3. 60 m is 3 cm on the map.

Part (b)

1. On the map the table must be more than 3 cm from P and less than 100\div20=5 cm from G.
2. W is exactly 3 cm from P, which is not more than 3 cm. Z is only 2 cm from P.
3. Y is exactly 5 cm from G, which is not less than 5 cm.
4. X is 5 cm from P and 2 cm from G, so it meets both conditions. The answer is X.