• Skip to primary navigation
  • Skip to main content
  • Skip to primary sidebar
Geometry Help
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
menu icon
go to homepage
search icon
Homepage link
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
×
Home » Circles » Chords » Chords that Bisect Each Other

Chords that Bisect Each Other

Last updated: Oct 9, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In this post, we will show that chords that bisect each other are diameters of the circle.

Problem

In circle O, chord AB bisects chord CD, and chord CD bisects chord AB. Prove that AB and CD are both diameters of the circle.

bisecting chords.

Strategy

One way to show that a chord is a diameter is to show that the two arcs formed by the chords are equal. Another way is to show that one of the arcs is subtended by a central angle measuring 180°, or equivalently, that it is subtended by an inscribed angle that measures 90° (since the inscribed angle's measure is half that of the central angle of the same arc)

In this problem, we don't know anything about the arcs, so let's see if we can prove it using inscribed angles.

We can connect the edges of the chords, forming a quadrilateral, and 4 triangles. Since the chords bisect each other, we have two pairs of equal halves of the chords, and it is easy to show that the two pairs of triangles are congruent using SAS. Then, the opposite sides of the quadrilateral are equal as corresponding sides in congruent triangles: AD=BC and DB=AC.

Bisecting chords form congruent triangles.

it follows that AD+DB = BC+AC, and the arcs of equal chord are equal, so 
arc(AD)+arc(DB)= arc(BC)+arc(AC), and so arc(ADB) = arc(BCA).

Chord BA thus separates the circle into two equal arcs, so it is a diameter, and we can similarly show the arcs formed by CD are equal which means it is a diameter, too.

Proof

Here's how you prove that chords that bisect each other are diameters:

(1) AO=OB //Given
(2) CO=OD //Given
(3) ∠α≅∠β  //Vertical angles
(4) △OAD ≅△OCB //(1), (2), (3), Side-Angle-Side postulate
(5) AD=CB //(4) corresponding sides of congruent triangles (CPCTC)
(6) ∠AOC ≅ ∠BOD //Vertical angles
(7) △AOC ≅△BOD //(1), (2), (6), Side-Angle-Side postulate
(8) AC=DB //(7) corresponding sides of congruent triangles (CPCTC)
(9) Arc(AC) = Arc(DB) //(8), equal chords have equal arcs
(10) Arc(AD) = Arc(CB) //(5), equal chords have equal arcs
(11) Arc(AD)+Arc(DB)= Arc(BC)+Arc(AC) //(9), (10) Additive Property of Equality
(12) Arc(ADB) = Arc(BCA) //(11)
(13) AB is a diameter //(12), diameter bisects circle into equal arcs

« Equal Chords Have Equal Arcs

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

Primary Sidebar

About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

By accessing or using this website, you agree to abide by the Terms of Service and Privacy Policy.


Copyright © 2023