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Home » Circles » Chords » A Diameter Perpendicular to a Chord

A Diameter Perpendicular to a Chord

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

We've shown that a diameter that bisects a chord is perpendicular to that chord. Here, we will prove the converse theorem.

Problem

In circle O, the diameter AB is perpendicular to a chord CD. Prove that it bisects the chord.

Geometry drawing of a chord and bisectors

Strategy

We will show that the two line segments, CE and ED, are equal, using triangle congruency. To do that we will construct triangles in which these two line segments are sides.

We can connect the chord's endpoints with the center. The connecting lines are radii of the circle, and thus equal to each other., since all the radii of a circle are equal. We will also use the fact that AB⊥CD (given in the problem statement) to prove congruency of these 2 right triangles using the Hypotenuse -Leg postulate.

Geometry drawing of a bisected chord with triangles.

Proof

(1) OE=OE // Common side, reflexive property of equality
(2) OC=OD=r // Radii of a circle are all equal
(3) AB⊥CD // Given
(4) ΔOEC≅ ΔOED // Hypotenuse-Leg
(5) CE=ED // (4), Corresponding sides in congruent triangles

And as a side benefit, we have also proven that the perpendicular diameter bisects the arc subtended by the chord, since:
(6) ∠COB ≅ ∠DOB // (4), Corresponding angles in congruent triangles
(7) ArcCB≅ArcBD //(6)

« A Diameter Bisecting a Chord
A Diameter that Bisects Two Chords »

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
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    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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