Construction: Perpendicular Bisector

Let's say I have a circle $\omega$ with center $O$.

And a chord $AB$.

Now make a perpendicular line from $O$ to $AB$

You've guessed it! If we have a chord $AB$ in a circle, and we draw a perpendicular line from $O$ to the chord, then this perpendicular line bisects $AB$

But is the converse true?

So how do we prove the first statement? Think about it; it's actually quite simple :)

Because $OA = OB$ and $OC = OC$, we have $\triangle AOC \cong \triangle BOC$ by HL congruence. Now, try to prove the converse on your own!

While we solve geometry problems, it is very pertinent to draw a diagram so that we can make up our mind about what does the question want. In this article we are going to talk about how to construct a perpendicular line.

Prior to start the messy work (maybe it's not that messy?), you need: a compass, a ruler and of course, a paper.

Steps: Start with an arbitrary line AB,

Grab a compass, then draw a circle centered at A, and followed by a circle centered at B (radii of both circle must be the same and longer than half of the line), and you should get something like this.

Just ignore the point C, that point is not necessary. Now, connect the intersection points of the two circles, D and E in the diagram. You get your perpendicular bisector of AB! In fact the intersection point of AB and the perpendicular bisector is the midpoint of AB.

Example Walk Through: How to Construct a Perpendicular Bisector $\\$

Can we find the perpendicular bisector of a line segment without measuring to find the midpoint and right angle?

Let's start with line segment AB.

Use one end of the line segment as the center and make a circle (Let's start with A here).

Estimate and make sure the radius is bigger than half of the line segment.

Create another circle with the same radius, this time with B as the center.

You can also do the above step but create arcs, instead of circles.

Connect the intersection, and you have the perpendicular bisector!

If you want to see why this method works, read **Walk through: Why does the construction method work? **

Walk through: Why does the construction method work? $\\$

To see how to construct a perpendicular bisector, read Example Walk Through: How to Construct a Perpendicular Bisector.

Let's start from the beginning.

Connect the intersections of the arcs/circles, we get the perpendicular bisector.

Connect some more lines to help us out

Connect the two end points of the line segment and the intersections. The four new line segments are equal because they are all radii of the two circles (with the same radius!)

Now to show that $\overline{CD}$ is a perpendicular bisector of $\overline{AB}$, we show the two properties: perpendicular and bisector, separately.

First, let's show perpendicularity

This part is a bit tricky. We rely a lot on the fact that the sum of 3 angles in a triangle is 180 degrees.

Next, show that $\overline{CD}$ is a bisector of $\overline{AB}$