Points 3 cm from A Easy

The question

Question: draw the locus of the points that are 3 cm from a point A.

A locus is the set of all the points that follow a rule.

Here: exactly 3 cm from A.

1 Try some points

Some points 3 cm from A, in different directions.

They all lie on a circle.

2 Draw the locus

Open the compass to 3 cm and draw a full circle around A.

Every point on it is 3 cm from A.

Careful!

The locus is only the circle itself.

Points inside are less than 3 cm away.

3 Check

Let's check: measure from A to any point of the circle: 3 cm.

So it is right!

Let's look back.

First: try some points.

Then: draw the locus.

Last: check.

▶ Watch it solved

Points the same distance from A and B, AB = 5 cm Medium

The question

Question: draw the locus of the points that are the same distance from A and from B, with A B = 5 cm.

A locus is the set of all the points that follow a rule.

Here: the same distance from A as from B.

1 Draw AB

Draw A B, 5 cm long.

2 Try some points

The middle of A B is the same distance from both.

So is a point straight above the middle.

They all lie on one line.

3 Arcs from A

Open the compass to more than half of A B: 3.5 cm.

From A, draw an arc above and an arc below.

Careful!

Do not change the compass.

The arcs from B must have the same opening as the arcs from A.

4 Arcs from B

Keep the same opening.

From B, draw two arcs.

They cross the first arcs at P and Q.

5 Join the crossings

With a ruler, join P and Q.

This line cuts A B in half at M, at a right angle: the perpendicular bisector.

6 Check

Let's check: A M and M B are both 2.5 cm, and the angle at M is 90°.

So it is right!

Let's look back.

First: draw AB.

Then: try some points.

Then: arcs from A.

Then: arcs from B.

Then: join the crossings.

Last: check.

▶ Watch it solved

Points less than 3 cm from A and nearer to A than to B, AB = 5 cm Harder

The question

Question: shade the points that are less than 3 cm from A and nearer to A than to B, with A B = 5 cm.

Less than, and nearer: the edges themselves are not included.

So we draw them dashed.

1 Read the rules

A and B are 5 cm apart.

Two rules: near A, and nearer to A than to B.

2 Less than 3 cm from A

Points 3 cm from A make a circle.

Less than 3 cm: inside it.

Dashed, because the circle itself is not included.

3 Nearer to A than to B

Points the same distance from A and B are on the perpendicular bisector of A B.

Nearer to A: on A's side of it.

Careful!

The region must follow both rules.

Only the part that is inside the circle and on A's side.

4 Shade the region

Shade the part inside the circle, on A's side of the line.

5 Check

Let's check one point in the shaded part: it is 1.5 cm from A, less than 3 cm, and 6.5 cm from B, so nearer to A.

So it is right!

Let's look back.

First: read the rules.

Then: less than 3 cm from A.

Then: nearer to A than to B.

Then: shade the region.

Last: check.

▶ Watch it solved