bisector

The student draws the bisector of the angle with a ruler and compass.

Definition

1. Noun:
   • A line that divides something into two equal parts: In geometry, a "bisector" is a line or segment that splits an angle, line segment, or shape into two congruent (equal) halves.
   • Specifically, an angle bisector: A ray that divides an angle into two equal angles.
   • A segment bisector: A line, ray, or segment that passes through the midpoint of a given segment, dividing it into two equal lengths.

Usage Examples

• Noun:
  • The bisector of a 60-degree angle creates two 30-degree angles. (A line that splits an angle into two equal parts.)
  • In the triangle, the angle bisector from vertex A meets the opposite side at its midpoint. (A line that divides an angle equally within a triangle.)
  • The perpendicular bisector of a line segment is a line that cuts the segment into two equal halves at a right angle. (A line that both bisects and is perpendicular to the segment.)

Advanced Usage

• "Angle bisector theorem": A geometric theorem stating that the angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides.

  • Using the angle bisector theorem, we can find the length of the segment on the opposite side. (A rule applied to triangles.)

• "Perpendicular bisector": A line that is both perpendicular to a segment and passes through its midpoint.

  • The perpendicular bisector of a chord in a circle passes through the circle's center. (A specific type of bisector used in circle geometry.)

Variants and Related Words

• Bisect (verb): to divide into two equal parts.

  • The line bisects the angle perfectly. (The line cuts the angle into two equal halves.)

• Bisection (noun): the act or process of dividing into two equal parts.

  • The bisection of the segment was done with a compass and ruler. (The action of splitting something equally.)

• Bisector of an angle (phrase): a specific type of bisector used for angles.

  • The bisector of an angle is always inside the angle. (A geometric term.)

Synonyms

• Divider: something that splits into parts (less specific).
• Halver: a term sometimes used informally for something that halves an object (rare in geometry).
• Midpoint line: a descriptive phrase for a segment bisector.

Phrasal Verbs

• (None directly applicable, as "bisector" is a noun, but the verb "bisect" can be used in phrases like "bisect into" or "bisect through".)
  • The line bisects through the center of the circle. (The line cuts through the center, dividing the circle into equal halves.)

Related Idioms

• (No common idioms use "bisector," as it is a technical geometric term. However, the concept appears in phrases like "cut in half" or "split down the middle.")
  • The argument bisected the group into two equal factions. (Metaphorical use of the verb "bisect" to mean dividing evenly.)