c part

Asked by Kanwaranita10 | 3rd Mar, 2020, 05:22: PM

Expert Answer:

In the given figure, TP and TQ are two tangents to the circle with centre O, touching at A and C respectively. If angle BCQ = 55o and angle BAP = 60o, find:
(i) angle OBA and angle OBC
(ii) angle AOC
(iii) angle ATC
Solution:-
left parenthesis i right parenthesis
T P space a n d space T Q space a r e space tan g e n t s space a t space A space a n d space C
T h e r e f o r e comma space O C space perpendicular space T Q space a n d space O A space perpendicular space T P
therefore space angle O C T equals 90 degree equals angle O C Q space space a n d space space angle O A T equals 90 degree equals angle O A P
N o w comma space angle O C Q equals angle O C B plus angle B C Q
rightwards double arrow angle O C B equals angle O C Q minus angle B C Q equals 90 degree minus 55 degree equals 35 degree
A l s o comma space space angle O A P equals angle O A B plus angle B A P
rightwards double arrow angle O A B equals angle O A P minus angle B A P equals 90 degree minus 60 degree equals 30 degree
I n space triangle O A B comma space O A equals O B space... space open parentheses R a d i i space o f space s a m e space c i r c l e close parentheses
rightwards double arrow angle O B A equals angle O A B space... space open parentheses A n g l e s space o p p o s i t e space t o space e q u a l space s i d e s close parentheses
rightwards double arrow angle O B A equals 30 degree
I n space triangle O B C comma space O B equals O C space... space open parentheses R a d i i space o f space s a m e space c i r c l e close parentheses
rightwards double arrow angle O C B equals angle O B C space... space open parentheses A n g l e s space o p p o s i t e space t o space e q u a l space s i d e s close parentheses
rightwards double arrow angle O B C equals 35 degree
left parenthesis i i right parenthesis
therefore angle A B C equals angle O B A plus angle O B C equals 65 degree
A n g l e space a t space t h e space c e n t r e space i s space t w i c e space t h e space a n g l e space a t space r e m a i n i n g space c i r c u m f e r e n c e
rightwards double arrow angle A O C equals 2 angle A B C equals 130 degree space space space space
left parenthesis i i i right parenthesis
N o w comma space A O C T space i s space a space q u a d r i l a t e r a l space
W e space k n o w space t h a t space t h e space o p p o s i t e space a n g l e s space o f space a space q u a d r i l a t e r a l space a r e space s u p p l e m e n t a r y
therefore angle A O C plus angle A T C equals 180 degree
rightwards double arrow angle A T C equals 180 degree minus 130 degree equals 50 degree

Answered by Renu Varma | 4th Mar, 2020, 11:52: AM