Bisector: Definitions and Examples

Bisector: Definitions, Formulas, & Examples

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    Home / Educational Resources / Math Resources / Bisector: Definitions and Examples

    A bisector is a line, segment, or plane that divides an angle, line segment, or shape into two equal parts. In geometry, the term bisector is often used to refer to the angle bisector, which is a line or segment that cuts an angle into two congruent angles.

    Bisector: Real World Application

    1. Construction: Bisector lines are commonly used in construction and engineering to ensure that buildings, bridges, and other structures are properly aligned and symmetrical.
    2. Surveying: In land surveying, bisector lines are used to determine the exact location of property boundaries and to ensure that structures are built on the correct parcels of land.
    3. Geometry: Bisector lines are used in geometry to find the midpoint of a line segment, the center of a circle, and the angle bisector of a triangle.
    4. Robotics: Bisector lines are used in robotics and computer vision to locate and track objects, as well as to navigate and map out environments.
    5. Graphics and design: Bisector lines are used in graphics and design to create symmetrical and balanced compositions, as well as to align text and other elements in a layout.
    6. Photography: In photography, the rule of thirds is a composition rule that suggest to align a subject with the points of intersection of bisector lines.

    Definitions:

    • Angle bisector: A line or segment that cuts an angle into two congruent angles.
    • Line bisector: A line that divides a line segment into two congruent segments.
    • Median: A line segment that connects the midpoint of a side of a triangle to the opposite vertex.
    • Altitude: A line segment that connects a vertex of a triangle to the line containing the opposite side, such that it is perpendicular to the line.
    • Perpendicular bisector: A line or segment that is perpendicular to and bisects a line segment.

    Examples:

    1. A line bisector of a line segment is a line that passes through the midpoint of the segment and is parallel to the segment. For example, if the line segment has endpoints A and B, the line bisector would be a line that passes through the midpoint of AB and is parallel to AB.
    2. An angle bisector of an angle is a line or segment that cuts the angle into two congruent angles. For example, if angle AOB is bisected by line segment CD, then angle AOC is congruent to angle BOD.
    3. A median of a triangle is a line segment that connects the midpoint of a side of a triangle to the opposite vertex. For example, if triangle ABC has a median from vertex A to midpoint D of side BC, the median AD is a line segment connecting vertex A to the midpoint D of side BC.
    4. An altitude of a triangle is a line segment that connects a vertex of a triangle to the line containing the opposite side, such that it is perpendicular to the line. For example, if triangle ABC has an altitude from vertex A to line BC, the altitude AE is a line segment connecting vertex A to the line BC, such that it is perpendicular to line BC.
    5. A perpendicular bisector of a line segment is a line or segment that is perpendicular to and bisects the line segment. For example, if the line segment has endpoints A and B, the perpendicular bisector would be a line that passes through the midpoint of AB, is perpendicular to AB and bisects the line segment.

    10 Question Quiz:

    1. What is the definition of an angle bisector?
    2. What is the definition of a line bisector?
    3. What is the definition of a median?
    4. What is the definition of an altitude?
    5. What is the definition of a perpendicular bisector?
    6. What is the name of the line or segment that cuts an angle into two congruent angles?
    7. What is the name of the line that divides a line segment into two congruent segments?
    8. What is the name of the line segment that connects the midpoint of a side of a triangle to the opposite vertex?
    9. What is the name of the line segment that connects a vertex of a triangle to the line containing the opposite side, such that it is perpendicular to the line?
    10. What is the name of the line or segment that is perpendicular to and bisects a line segment?

    Answers:

    1. Angle bisector
    2. Line bisector
    3. Median
    4. Altitude
    5. Perpendicular bisector 6

    Bisector:

    Illustration

    Illustration

    Alternate names

    bisector | interior angle bisector | internal angle bisector

    Definition

    The (interior) bisector of an angle, also called the internal angle bisector, is the line or line segment that divides the angle into two equal parts. The angle bisectors meet at the incenter I, which has trilinear coordinates 1:1:1. The length t_1 of the bisector A_1 T_1 of angle A_1 in the above triangle Δ A_1 A_2 A_3 is given by t_1^2 = a_2 a_3[1 - a_1^2/(a_2 + a_3)^2], where t_i congruent (A_i T_i)^_ and a_i congruent (A_j A_k)^_. The points T_1, T_2, and T_3 have trilinear coordinates (0, 1, 1), (1, 0, 1), and (1, 1, 0), respectively, and form the vertices of the incentral triangle.

    Related terms

    angle | angle bisector theorem | angle trisection | cyclic quadrangle | exterior angle bisector | incenter | incentral triangle | incircle | isodynamic points | orthocentric system | Steiner-Lehmus theorem

    Related Wolfram Language symbol

    AngleBisector

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