Investigation 8: Geometric Construction of Bisectors

New Concept

The word bisect means "to cut into two equal parts."

What’s next

Next, you’ll use a compass and straightedge to construct the perpendicular bisector of a segment and the bisector of an angle.

Property

The word bisect means 'to cut into two equal parts.' A segment is a part of a line with two distinct endpoints and can be bisected. A line cannot be bisected because it has no midpoint.

Examples

A perpendicular bisector is a line that bisects a segment and is also perpendicular to it.

Step 1: To find the perpendicular bisector of segment $XY$, set your compass to a width more than half of $XY$'s length.
Step 2: Swing arcs from point $X$ above and below the segment, then repeat from point $Y$ without changing the compass width.
Step 3: Draw a straight line connecting the two points where the arcs intersect to form the perpendicular bisector.

Imagine a line segment is a tightrope. A perpendicular bisector is like a support pole that is placed exactly in the middle of the rope and stands perfectly straight up at a 90-degree angle. It's the ultimate balance point, providing perfect support by cutting the segment into two equal halves at a perfect right angle.

An angle bisector is a ray drawn halfway between the two sides of an angle, dividing it into two smaller angles of equal measure.

Step 1: To bisect $\angle ABC$, place your compass on the vertex $B$ and draw an arc that crosses both sides of the angle.
Step 2: From the points where the arc intersects the sides, swing two new arcs in the middle of the angle so they cross.
Step 3: Draw a ray from the vertex $B$ through the point where the two arcs intersect. This ray is the angle bisector.

Think of an angle as a perfect slice of pizza. The angle bisector is the cut you make right down the middle, from the pointy tip to the crust, that divides your slice into two smaller, identical slices. This ensures that you and a friend get a fair share of that delicious pizza angle. Everybody's happy!