CBSE Class 9 Answered

Prove that the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Asked by mkmishra101 | 03 Mar, 2016, 05:38: PM

Expert Answer

An angle can be subtended by

1. Minor arc

2. Major arc

3. The semi circle

begin mathsize 14px style 1. space Minor space arc. In space figure space 1 space given space below comma space In space increment space OCB comma space angle space DOB space equals space angle OCB plus angle OBC space horizontal ellipsis left parenthesis 1 right parenthesis space left parenthesis A n space e x t e r i o r space a n g l e space o f space t r i a n g l e space i s space e q u a l space t o space t h e space s u m space o f space i n t e r i o r space a n g l e s. right parenthesis space OB equals space OC space horizontal ellipsis left parenthesis space radius space of space circle right parenthesis space angle OCB equals space angle OBC space space open parentheses angles space opposite space to space equal space sides close parentheses therefore angle DOB space equals space 2 angle OCB space space horizontal ellipsis left parenthesis 2 right parenthesis In space increment space OCA space angle DOA space equals space OCA plus OAC space horizontal ellipsis left parenthesis 3 right parenthesis space OC equals space OA space horizontal ellipsis left parenthesis space radius space of space circle right parenthesis space angle OCA equals space angle OAC space space open parentheses angles space opposite space to space equal space sides close parentheses therefore angle DOA space equals space 2 angle OCA space space horizontal ellipsis left parenthesis 4 right parenthesis Hence comma space from left parenthesis 2 right parenthesis space and left parenthesis 4 right parenthesis space angle DOB space plus angle DOA equals 2 angle OCB space space plus space 2 angle OCA therefore angle AOB equals 2 open parentheses angle OCB plus angle OCA close parentheses equals 2 angle ACB Hence comma space proved. end style

Similarly this can be proved for major arc (figure 2) and semi-circle (figure 3).

Answered by Priti Shah | 03 Mar, 2016, 07:30: PM

CBSE Class 9 Answered

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