Equal Chords of a Circle Subtend Equal Angle at the Centre


 
 
Concept Explanation
 

Equal Chords of a Circle Subtend Equal Angle at the Centre

Theorem 1   Equal chords of a circle subtend equal angles at the centre.

Given    A circle C (O,r) and its two equal chords AB and CD.

TO PROVE   large angle AOB =angle COD

PROOF     In triangles AOB and COD, we have

                      AB = CD     [ Given ]

                     OA = OC      [ Each equal to r ]

                     OB = OD      [ Each equal to r ]

So, by SSS- criterion of congruence, we have

                    large Delta AOB cong Delta COD

large Rightarrow                    large angle AOB = angle COD                             [C.P.C.T.]

 

Theorem 2   IF the angles subtended by two chords of a circle at the centre are equal , the chords are equal.

Given      Two chords AB and CD of a circle C (O,r) such that

                      large angle AOB = angle COD.

TO PROVE       AB = CD

PROOF      In triangles AOB and COD, we have

                       large OA = OC                       [ Each equal to r]

                  large angle AOB = angle COD      [ Given ]

                      large OB = OD                       [ Each equal to r ]

So, by SAS- criterion of congruence, we have

                    large Delta AOB cong Delta COD

large Rightarrow                    large AB = CD                      [C.P.C.T.]

Illustration: Inthe given figure AB = PQ and angle AOB= 65^0

Solution:  in the figure it  is given AB = PQ

We know that equal Chords subtends equal angle at the centre

angle AOB= 65^0                    [Given]

and angle AOB= angle POQ

therefore angle POQ= 65^0

 

 

Sample Questions
(More Questions for each concept available in Login)
Question : 1
In the given figure, O is the centre of the circle, AM=PQ;and ; angle POQ=45^o. The measure of angle AOM is

 

Right Option : A
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Question : 2

If the line segment joining P and Q is equal to line segment joining R and S, then which of the following statements is true ?

 

Right Option : B
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Explanation
Question : 3

In the circle given below, O is the centre and chord AM = chord PQ. If  angle AOM=60^o, then angle POQ is equal to

          

 

Right Option : B
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Explanation
 
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