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Question

BC is a chord with centre O. A is a point on an arc BC as shown in the figure. Prove that:

(i) BAC+OBC=90, if A is the point on the major arc

(ii) BACOBC=90, if A is the point on the minor arc.
[4 MARKS]


Solution

Each subpart: 2 Marks each

(i) We observe that the minor arc BC makes BOC=2BAC at the centre 

Let BAC=x

z=2x ...... (1)

In OBC, we have

OBC+OCB+BOC=180

y+z+y=180

2x+2y=180  [From (1)]

x+y=90

BAC+OBC=90


(ii) We observe that the major arc BC subtends BOC=z and O=t, at the centre and BAC=x, at a point on the circumference.

z=2x

In ΔOBC, we have

OBC+OCB+BOC=180

y+y+t=180

t=1802y

Now, z=360t

z=360180+2y

2x=180+2y [z=2x]

xy=90

BACOBC=90

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