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Home » Circles » Chords » A Diameter that Bisects Two Chords

A Diameter that Bisects Two Chords

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will prove that if a diameter bisects two chords in a circle, the two chords are parallel to each other. We can do this this three ways, relying on the properties of parallel and perpendicular lines.

Problem

In circle O, AB is a diameter that bisects two chords, CD and EF. Show that the chords are parallel to each other.

Geometry drawing of two parallel chords

Strategy

To show two lines are parallel, we can use one of the several methods: either show congruent corresponding angles, congruent alternating interior angles, or the Perpendicular Transversal Theorem.

Since we have shown that the diameter that bisects a chord is perpendicular to that chord. any of these three methods will work, and all are as easy to show. We'll prove this using each of the three.

Proof

Here's how you prove that if a diameter bisects two chords, they are parallel to each other:

(1) AB bisects CD // Given
(2) AB⊥CD // A diameter that bisects a chord is perpendicular to that chord
(3) AB bisects EF // Given
(4) AB⊥EF // A diameter that bisects a chord is perpendicular to that chord
(5) ∠OGD ≅∠AHF=90° //(2), (4), definition of perpendicular lines
(6) EF||CD // congruent corresponding angles

Alternatively, we can repeat the first part of the proof, above, through step number (4), to show that AB is parallel to both CD and EF (AB⊥CD and AB⊥EF), and from there proceed as follows:
(5)' ∠OHF ≅∠OGC=90° //(2), (4), definition of perpendicular lines
(6)' EF||CD //Converse congruent alternating interior angles theorem

Finally, to prove this a third way, using the perpendicular transversal theorem, we again repeat the first part of the proof, above, through step number (4). This shows that AB is perpendicular to both CD and EF (AB⊥CD and AB⊥EF), and from there proceed as follows:

(5)'' EF||CD //(2),(4) Perpendicular Transversal Theorem.

« A Diameter Perpendicular to a Chord
Congruent Chords are Equidistant from the Center »

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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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

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