Points the same distance from the two arms of an angle of 60°

Step by step

The question

Question: draw the locus of the points that are the same distance from the two arms of an angle of 60°.

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

Here: the same distance from both arms.

The arms are the two sides of the angle, meeting at V.

The distance to a line is measured straight across, at a right angle.

Fold the paper along the line that cuts the angle in half: one arm lands on the other.

So each point on the fold is the same distance from both arms.

We construct that line.

1 Draw the angle

Draw the two arms meeting at V, with 60° between them.

2 Arc from the corner

From the corner V, draw an arc that cuts both lines, at P and Q.

3 Arcs from P and Q

Open the compass to any size.

With that same opening, draw an arc from P and an arc from Q, inside the angle.

They cross at R.

Careful!

Keep the compass the same for both arcs.

Then R is the same distance from P and from Q.

4 Join V and R

Join V to R with a ruler, and make the line longer.

V P and V Q are the same length: one arc made both.

P R and Q R are the same length too.

So if we fold the paper along V R, P lands on Q, and the two halves of the angle fit exactly.

So V R cuts the angle in half.

Each half is 60 divided by 2: 30°.

5 Check

Let's check with a protractor: both halves are 30°.

So it is right!

Let's look back.

First: draw the angle.

Then: arc from the corner.

Then: arcs from P and Q.

Then: join V and R.

Last: check.