Question

•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​

Answer 1

$\large\underline{\sf{Solution-}}$

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

$\rm \: \angle 1 + \angle 2 = 2\angle 4 + 2\angle 5$

$\rm \: \angle 1 + \angle 2 = 2(\angle 4 + \angle 5)$

$\rm\implies \:\boxed{ \rm{ \:\angle AOB \: = \: 2\angle ACB \: \: }}$

$\rule{190pt}{2pt}$

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.

Answer 2

Step-by-step explanation:

Answer:

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Step-by-step explanation:

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