A circle with centre 'O' has a radius √2 cm, it is divided into two segments by a chord AB of length 2 cm. Prove that the angle subtended by the chord at a point 'P' in the major segment is 450.
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Solution
Given radius =√2cm
Therefore AO=√2cm
Let OD be the perpendicular from O on AB
And AB =2cm
Therefore AD=1cm (perpendicular from the centre bisects the chord)
Now in triangle AOD,
AO=√2cm
AD=1cm
And let angle AOD =θ
Therefore , sinθ=1√2
Hence, θ=45o
Therefore angle AOB =45o+45o=90o
Then angle APB =902=45o {angle made by a chord at the centre is double of the angle made by the chord at any poin on the circumference)