Mathematics
In the figure (i) given below, C and D are centers of two intersecting circles. The line APQB is perpendicular to the line of centers CD. Prove that
(i) AP = QB
(ii) AQ = BP.
Circles
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Answer
Let M be the point of intersection of line CD and line APQB.
(i) Since, the perpendicular to a chord from the centre of the circle bisects the chord,
∴ In circle with center D,
PM = MQ ………(1)
and
In circle with center C,
AM = MB ………(2)
Subtracting equation (1) from (2),
⇒ AM - PM = MB - MQ
⇒ AP = QB
Hence, proved that AP = QB.
(ii) Let AP = QB = x.
From figure,
AQ = AB - QB = AB - x
BP = AB - AP = AB - x.
∴ AQ = BP.
Hence, proved that AQ = BP.
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