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Angle at the Centre is Double The Angle at Circumference

Circles III - Concepts

Angle at the Centre is Double The Angle at Circumference

Angles in the Same Segment are Equal

Angle in a Semicircle

Cyclic Quadrilateral

Class - 9th IMO Subjects

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 $large large angle OGK=x^0$ .

$Delta OGH$ is an isosceles. LengthsOH and OG are both radii of the same circle

$angle OGH=angle OHG = y^0$ [

$Delta OGK$ is 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 $angle ABC;?$

Image required

$45^o$

$90^o$

$135^o$

$270^o$

Right Option : A

View 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

in the interior of S

in the exterior of S

on the segment AB

on AB and bisects AB

Right Option : A

View Explanation

Question : 3

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

$30^o$

$45^o$

$60^o$

$90^o$

Right Option : A

View Explanation

Video Link - Have a look !!!

Language - English

Chapters

Polynomials II

Direct and Inverse Variation

Linear Equations in One Variable

Introduction to Trigonometry

Introduction to Euclids Geometry

Herons Formula Complete

Polynomial I

Surface Area and Volume II

Surface Area and Volume I

Coordinate Geometry

Constructions

Areas Of Parallelograms And Triangles II

Areas Of Parallelograms And Triangles I

Linear Equations in Two Variables

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Angle at the Centre is Double The Angle at Circumference

Angle at the Centre is Double The Angle at Circumference

Students / Parents Reviews [10]

Prisha Gupta

Shreya Shrivastava

Manish Kumar

Om Umang

Aman Kumar Shrivastava

Cheshta

Archit Segal

Barkha Arora

Eshan Arora

Shivam Rana