Perpendicular Bisectors

What is a perpendicular bisector?
Look at the words by themselves, first of all:

• Perpendicular: This means 'at right-angles to.' For example, walls are usually perpendicular to the floor, and table legs are usually perpendicular to the table top.
• Bisector: This means 'something that cuts exactly in half.' For example, if you snap a KitKat finger and the two pieces you have left over are exactly the same size, you have bisected it.

When you're asked to construct a perpendicular bisector, you'll be expected to draw a line that cuts another line exactly in half. Sometimes you'll be given two points and asked to find where something can go if it needs to be the same distance between them. Just pretend that the two points are opposite ends of a line* and use the same method as outlined below.


Tools
Wait! Before you start, you'll need the following tools:

• Pencil (diagrams must be drawn in pencil if you're working towards an exam, and your pencil needs to be well sharpened);
• A ruler or other straight edge;
• A pair of compasses**;
• A rubber (eraser) in case you make mistakes, or to rub out construction lines.
  

How to construct a perpendicular bisector
I'll be assuming that you've already got a line to bisect. If not, now's the time to draw one. Draw any line you like, but if you need a suggestion, go for one that's about 10cm long and draw it in the middle of your page to give yourself plenty of space. Now follow these steps:

Step 1.
Open up your compasses. It doesn't matter how wide, as long as it's at least half the distance between the two ends of your line.

Step 2.
Place the point of your compasses on one end of the line that you're trying to bisect; it doesn't matter which end. Now draw a circle with your compasses in the usual way.

Step 3.
DON'T CHANGE THE SIZE OF YOUR COMPASSES! <--- This bit is really, really, really important. It won't work if you close your compasses or open it further, and you'll have to start again.
Place the point of your compasses on the other end of your line, and draw a circle in the same way.

• If you've done it correctly, you should have two circles that overlap and are exactly the same size. If they're not the same size, or they don't overlap, you'll need to start again and see what you've missed out from these instructions!

Step 4.
Use your ruler to help you join up the two places where the circles cross over with a straight line. Extend your line as far as you can on either side of the diagram

• You've drawn your perpendicular bisector! The next bit is just about tidying up.

Step 5.
You might want to use your rubber to get rid of the circles and make it easier to see the result. Be warned though, that many exam questions will ask you to leave your construction lines visible - in this case, your circles are your construction lines and you should not rub them out!

Now take a look at the perpendicular bisectors example question to see a 'real life' example.


More information
When you become more confident at drawing this type of construction, it'll become clearer that it's not necessary to draw the whole of each circle in order to find the perpendicular bisector. With a bit of practise you'll get used to how much of the circle (or what size arc) you need to draw to get things done.

Other uses
The method for drawing a perpendicular bisector can also be used to:

• Find the midpoint of a line
• Find the midpoint between two points (draw a line between the two points, then find the midpoint of that)
• Find the locus of points equidistant from two given points
  

* Strictly speaking, a line segment.
** That's one set of the pointy type of compass, not two of the magnetic type. If you'd like help setting up your compasses, please comment and we'll sort out a post for it!