Equal Chords of a Circle

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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
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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
Cyclic Quadrilaterals
Tangent to Circle

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