Angle at the Centre is Double The Angle at Circumference


 
 
Concept Explanation
 

Angle at the Centre is Double The Angle at Circumference

Angle at the centre is double The Angle at circumference.

The angle subtended by an arc at the centre is twice the angle subtended at the circumference

The angle at the centre is double the angle at the circumference.

Given : Given a Circle with centre O and a Chord HK

To Prove large angle HOK=2angle HGK

Construction: Join OG, OH and OK. Extend HO to J 

Proof  Let large angle OGH=y^0 and large large angle OGK=x^0.

   Delta OGHis an isosceles. LengthsOH and OG are both radii of the same circle

angle OGH=angle OHG = y^0  [

Delta OGKis an isosceles. Lengths  OG and OK are both radii of the same circle.

angle OGK=angle OKG = x^0  [

large angle GOH=180-2y                  [ Angle sum property ]   ..............(1)

large angle GOK=180-2x                  [ Angle sum property ] ..............(2)

large angle HOK + angle GOK+angle GOH=360^0          [Angles around a point]                                                                                            ..............(3)

Putting the values from (1) and (2) in (3) we Get

large angle HOK +180-2x^0+180-2y^0=360^0

large angle HOK +360-2x^0 -2y^0=360^0

large angle HOK =360^0-360+2x^0 +2y^0

large angle HOK =2(x^0 +y^0)            ..............(4)      

large angle HGK =angle HGO+ angle OGK= x^0 +y^0   ..............(5)

From (4) and (5 ) we Get

large angle HOK =2angle HGK

Hence Proved.

Illustration:   If A, B, M are three points on a circle with centre O such that angle AOM=70^0 and  angle MOB=140^o, Find   angle AMB

 Solution:   If  A, B, M are three points on a circle with centre O such that angle AOM=70^0 and  angle MOB=140^o, 

 angle AOB = angle AOM + angle MOB = 70^0 +140^0 = 210^0

 Reflex of angle AOB=360^0- 210^0=150^0

angle AMB=frac{150^0}{2}=75^0

As angle at the centre is double the angle and the circumference.

Sample Questions
(More Questions for each concept available in Login)
Question : 1

O is the centre of the given circle. If reflex anglr AOC=270^o, which of the following is the measure of angle ABC;?

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Right Option : A
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Explanation
Question : 2
Let B be the mid-point of an arc AC of a circle such that m  angle AOB=183^o. If the region bounded by the arc ABC and line segment AC is denoted by S, then the centre O of the circle lies

 

Right Option : A
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Explanation
Question : 3

The value of large angle ABC in the following figure, where O is the centre of the circle, and large angle AOC islarge 60^0

 

Right Option : A
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Explanation
 
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