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Geometry: Constructions

Dividing Angles and Segments

Problems

Problems

Dividing Angles

Angles can be divided just like ordinary numbers. An angle can only be divided by a ray on the interior of the angle, though. Such a ray that divides an angle into two equal angles is called an angle bisector. Likewise, two rays that divide an angle into three congruent angles are called angle trisectors.

Figure %: An angle bisected and trisected
On the left, angle ABC is bisected by the ray BD. To know this, we must know that angle ABD and angle CBD are congruent. On the right, angle ABC is trisected by ray BE and ray BF. In this case, the three angles ABE, EBF, and FBC are congruent.

With angle bisectors and trisectors, it also holds true that any of the new angles created by the bisector or trisector is equal to exactly one-half or one-third of the original angle, depending on whether the angle has been bisected or trisected.

Dividing Segments

A segment is divided into two equal segments only when a line or segment passes through the midpoint of the original segment. The midpoint of a segment is the point lying in the segment that is exactly halfway from each endpoint of the segment.

Figure %: The midpoint of a segment
In the above figure, the segment AB is divided into two segments, AM and MB. Point M is the midpoint of segment AB, thus AM and MB are of the same length: one-half the length of AB.

Bisectors

When a line or segment passes through the midpoint of another segment, that line or segment is a bisector of the other segment. There are an infinite number of bisectors for every segment, depending on the angle at which the incoming segment or line bisects the other segment.

Figure %: A segment being bisected by many different lines and segments
The segment AB, with midpoint M, is bisected by segment CD, line EF, and segment GH.

Perpendicular Bisectors

If a bisector is perpendicular to the segment it bisects, it is called the perpendicular bisector of that segment. Because there exists only one line perpendicular to a line at a given point, a segment has only one perpendicular bisector: the perpendicular line that passes through the midpoint of the segment.

Figure %: A perpendicular bisector
The line CD contains the midpoint of segment AB, and forms a right angle with the segment. Therefore, it is the perpendicular bisector of segment AB.

Just as there are bisectors for segments, there are trisectors, too. Segment trisectors divide a segment into three equal segments.

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