Bisect

High School

Definition

To split a geometric shape into two congruent parts or exactly in half of equal shape and sizes. The line used to split the congruent shapes is called a "bisector" and the object being split is known as the "bisected." It is also possible to bisect angles and lines or segments. When bisecting an angle, the bisector passes through the apex of the angle splitting it into two equal angles. In three dimentional objects, bisection is done through planes, known as "bisecting plane."

Worked examples

\(\overline{AB}\) is bisected at point \(M\), so \(AM = MB\)

The bisector creates two segments of equal length; \(M\) is the midpoint.

\(\angle ABC = 80^\circ\) bisected by \(\overrightarrow{BD}\) gives \(\angle ABD = \angle DBC = 40^\circ\)

The angle bisector splits the original angle into two congruent angles of half the measure.

A perpendicular bisector of \(\overline{PQ}\) is \(\perp\) to \(\overline{PQ}\) at its midpoint

This special bisector cuts the segment in half and forms right angles with it.

Common mistakes

Bisecting \(\overline{AB}\) means cutting it into any two pieces → Bisecting \(\overline{AB}\) means cutting it into two congruent (equal-length) pieces Bisect always means exactly in half, not just any split.
The bisector of \(\angle ABC\) passes through point \(B\) and any point inside → The angle bisector passes through vertex \(B\) and splits \(\angle ABC\) into two equal angles The bisector must create two congruent angles, not just any ray through the vertex.
A line bisects a circle if it passes through the center → A line (diameter) bisects a circle into two congruent semicircles if it passes through the center Correct idea, but be precise: the line divides the circle into two equal halves.

Where you'll use it next

You'll use bisectors when constructing perpendicular bisectors and angle bisectors with compass and straightedge, proving triangle congruence, finding midpoints and centroids, and later in coordinate geometry and trigonometry.

Found in 1 StudyPug lesson

Mastering Angle Bisectors: From Theory to Practice

Grade 7 Math

Dive into the world of angle bisectors! Discover construction techniques, explore real-world applications, and enhance your problem-solving skills with our comprehensive guide.

7th Grade7th