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Question

In the given figure, a square is inscribed in a circle with centre O.
Find:
OCB
Is BD a diameter of the circle?
1841216_2f98176fd6ca4b1ba6e6ffe76d1e2cf1.png

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Solution

In the given figure we can extend the straight line OB to BD and CO to CA
Then we get the diagonal of the square which intersect each other at 90o by the property of Square.
From the above statement we can see that
COD=90o,
The sum of the angle BOC and COD is 180o as BD is a straight line.
Hence BOC+OCD=BOD=180o
BOC+90o=180o
BOC=180o90o
BOC=90o
Thus, triangle OCB is an isosceles triangle with sides OB and OC of equal length as they are the radii of the same circle.
in OCB,OBC=OCB as they are opposite angles to the two equal sides of an isosceles triangle.
Sum of all the angles of a triangle is 180o
so, OBC+OCB+BOC=180o
OBC+OBC+90o180o as, OBC=OCB
2OBC=180o90o
2OBC=90o
OBC=45o
as OBC=OCB So,
OBC=OCB=45o
Yes BD is the diameter of the circle.
1799411_1841216_ans_172abc0b03d34a7f9b4869f87454a5e2.png

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