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What are compass and straightedge constructions?

A (mathematical) straightedge is thought of as an infinitely long ruler with no markings - so that you can connect any two points with a straight line, but you can't measure distances numerically.

A (mathematical) compass is a tool for constructing circles of any radius centered at any chosen point.

There are two types of compasses: collapsible and non-collapsible, the latter of which you can use to copy a radius from one center to another as well, without the compass "collapsing". It turns out that it doesn't matter whether we use a collapsible or non-collapsible compass as far as what things we can construct, so for simplicity we'll assume our compass is non-collapsible.

Now, let's try to construct the perpendicular bisector of a segment using only a straightedge and compass.

Start with a segment AB:

There's not much we can do here except draw the circle centered at A with radius B and draw the circle centered at B with radius A. Let's draw them both:

We can use our straightedge to draw a line through the two points of intersection of the two circles:

The perpendicular bisector of AB is the locus of all points equidistant from A and B, and the two points C and D are equidistant because they are the radii of congruent circles! Therefore we can construct perpendicular bisectors using a straightedge and compass.

Introduction to constructions: What is it and Why is it unique? $\\$

Constructions are drawing different kinds of lines and shapes in geometry with a compass and a straightedge (or ruler), without the measurement of lengths or angles.

Compasses are used to draw arcs and circles, straightedges or rulers are used to draw lines and straight angles.

A Little Bit of History$\\$ Construction is usually associated with Euclidean Geometry, named after the famous Greek mathematician Euclid. In his book "Elements", Euclid postulates what we know as algebra and number theory in geometrical language; he also uses a lot of construction techniques. If you're interested, read more about Euclidean Geometry here.

Measuring vs. Constructing$\\$ There are different ways to say, find the perpendicular bisector of a line. Here, we will show how to do it in Euclidean Geometry with construction; and how to do it in Analytical Geometry with coordinates and measurements.

Through constructing $\\$ Start with line segment AB.

Make two circles or arcs with the same radius, one with A as the center and the other with B. Connect the two intersections of the arcs, and you have the perpendicular bisector!

Through measuring $\\$ Start with the same line segment AB, but this time on a coordinate plane (makes measuring a bit easier, you could also use a ruler to measure).

Find the mid-point via measuring and make a perpendicular that goes through that point.