•Relation between Angle at centre and angle at the circumference The angle subtended at the centre of a circle by an arc is double the angle subtended on the circumference by the same arc. Created by amishafilomeena1003.

•Relation between Angle at centre and angle at the circumference The angle subtended at the centre of a circle by an arc is double

[tex]\large\underline{\sf{Solution-}}[/tex]

Given :- A circle with center O, arc AB of a circle subtends ∠AOB at the center and ∠ACB on the circumference of the circle.

To Prove :- ∠AOB = 2∠ACB

Construction :- Join CO and produced to D

In general three cases arises.

Case :- 1 Angle subtended by minor arc [ Figure 1 ]

Case :- 2 Angle subtended by semi arc [ Figure 2 ]

Case :- 3 Angle subtended by major arc [ Figure 3 ]

Proof :- [ All the three cases have the same proof ]

In ∆ AOC

OA = OC [ Radius of same circle ]

∴ ∠3 = ∠4 [ Angle opposite to equal sides are equal ]

Now, ∠1 is exterior angle of ∆ AOB

So, by exterior angle property, Exterior angle of a triangle is equals to sum of interior opposite angles.

⇛ ∠1 = ∠3 + ∠4

⇛∠1 = 2∠4 -----[ 1 ]

Similarly, ∠2 = 2∠5 ----- [ 2 ]

On adding equation (1) and (2), we get

[tex]\rm \: \angle 1 + \angle 2 = 2\angle 4 + 2\angle 5 \\ [/tex]

[tex]\rm \: \angle 1 + \angle 2 = 2(\angle 4 + \angle 5) \\ [/tex]

[tex]\rm\implies \:\boxed{ \rm{ \:\angle AOB \: = \: 2\angle ACB \: \: }} \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information :-

1. Angle in semi-circle is right angle.

2. Angle in same segments are equal.

3. Sum of the opposite pair of angles of cyclic quadrilateral is supplementary.

4. Exterior angle of a cyclic quadrilateral is equals to interior opposite angle.

5. Equal chords subtends equal angles at the center.

6. Equal chords are equidistant from the center.