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Home » Circles » Chords » A Diameter Bisecting a Chord

A Diameter Bisecting a Chord

Last updated: Oct 1, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

There are several theorems related to chords and radii or diameters that connect to them. In today's lesson, we will first prove that a diameter that bisects a chord is perpendicular to that chord and that it also bisects the arc that the chord subtends.

Problem

In circle O, the diameter AB bisects a chord, CD. Prove that AB⊥CD and that the two arcs, CB and BD are equal.

Geometry shapes: Chords and bisectors

Strategy

In order to prove that two lines are perpendicular, we can use the Linear Pair Perpendicular Theorem. This theorem states that if two straight lines intersect at a point and form a linear pair of equal angles, they are perpendicular.

AB and CD intersect at point E, so we'd need to prove the angles created at that intersection point, ∠OEC and ∠OED, are equal. That is easy to do with triangle congruency, if we connect the chord's endpoints with the center, as the sides are all equal.

This will also demonstrate that the two central angles that subtend the two parts of the arc CBD are equal, proving the second part.

Geometry drawing of a bisected chord with triangles.

Proof

(1) CE=ED //Given, AB bisects the chord CD
(2) OE=OE // Common side, reflexive property of equality
(3) OC=OD=r // Radii of a circle are all equal
(4) ΔOEC≅ ΔOED // Side-Side-Side postulate
(5) ∠OEC ≅ ∠OED // (4), Corresponding angles in congruent triangles
(6) AB⊥CD // Linear Pair Perpendicular Theorem
(7) ∠COB ≅ ∠DOB // (4), Corresponding angles in congruent triangles
(8) ArcCB≅ArcBD //(7)

Related Proofs

Now that we have our proof, we can use the same procedure to prove the converse, that a diameter that is perpendicular to a chord bisects that chord.

And we can also now easily show that if a diameter bisects two chords, they are parallel to each other.

« Radius of a Circle with an Inscribed Triangle
A Diameter Perpendicular to a Chord »

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]

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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
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    • Perpendicular lines
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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