# Equal Chords of a Circle

We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

Some Results on Equal Chords of a Circle
1) In a circle equal-chords are equidistant from the center. If AB = CD then OM = OL

Converse of the above result is also true.
If OM = OL then AB = CD

2) Equal-chords of congruent circles are equidistant from the corresponding centers. If two circles are congruent and AB = CD then OL = PM.

Converse of the above result is also true.
If two circles are congruent and OL = PM then AB = CD.
3)In a circle equal chords subtend equal angles at the center. In a circle, if AB = CD then ∠AOB = ∠COD

Converse of the above result is also true.
In a circle, if ∠AOB = ∠COD then AB = CD.

Some solved examples on the above result:

1) If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.

Given : ∠OAL = ∠OAM.

Prove that : AB = AC

Construction : Draw OL ⊥ AB and OM ⊥ AC Statements Reasons 1)OL ⊥ AB and OM ⊥ AC 1) By construction 2)∠OLA = ∠OMA 2)Each 900 3) OA = OA 3) Reflexive (common ) 4) ∠OAL = ∠OAM 4) Given 5) ΔOLA = ΔOMA 5) AAS postulate 6) OL = OM 6) CPCTC 7) AB = CD 7) Chords are equidistant from center O
_________________________________________________________________
2) If two equal-chords of a circle intersect within the circle, prove that the line joining the point of intersection to the center makes equal angles with the chords.

Given : PQ = RS They intersect each other at point T.

Prove that : ∠OTV = ∠OTU

Construction: Draw perpendiculars OV and OU on these chords. Statements Reasons 1) OV = OU 1) Equal-chords of a circle are equidistant from the center 2)∠OVT = ∠OUT 2)Each 90° 3) OT = OT 3) Reflexive (common) 4) ΔOVT ≅ ΔOUT 4) HL postulate or (RHS theorem) 5) ∠OTV = ∠OTU 5) CPCTC
Therefore, it is proved that the line joining the point of intersection to the center makes equal angles with the chords.

Circles

Circles
Parts of Circle
Arc and Chords
Equal Chords of a Circle
Arc and Angles