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Theorem 22

Theorem 2. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Given: - A circle with centre O and an arc AB subtending at the centre and on the remaining part of the circle at C.

To prove:-

Construction:- CO is joined and produced to P. OA and OB are joined

Proof:- In

OA = OC     (Radii)

[Angle opposite to equal sides]

[Exterior angle of ]

     [ ]

Similarity, by taking we have

Adding (i) and (ii) in fig (i) we have

Similarty, subtracting (i) from (ii) in fig. (ii) We have

Note: -

(i) Angle in a semi-circle is a right angle.
(ii) The circle drawn with hypotenuse of a right triangle as diameter pass through its opposite verdes or the arc of a circle subtending a right angle at any point on the remaining part of the circle is a semicircle.

 

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