Equal Chords of a Circle

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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