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Question

Which of the following is true, if AB=a,BC=b,CD=c and DA=d ?

A
a+b=c+d
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B
a+d=b+c
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C
a+c=b+d
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D
None of these
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Solution

The correct option is C a+c=b+d
A circle with centre O, touches the sides AB,BC,CD,DA, at P,Q,R,Srespectively.
Join OP,OQ,OR,OS
Now two tangents drawn from an external C
point to a circle subtend equal angles at
the centre.
(i)AP=AS, (ii)BP=BQ,(iii)CR=CQ, (iv)DR=DS
AP=AS (tangents from A) ....(*)
BP=BQ (tangents from B) ....(*)
CR=CQ (tangents from C) ....(*)
DR=DS (tangents from D) ....(*)
Adding the relations given in (*), we get
(AP+BP)+(CR+DR)=AS+BQ+CQ+DS
AB+CD=(AS+DS)+(BQ+CQ)
AB+CD=AD+BC
a+c=d+b
Hence choice (C) is correct answer.
365965_165105_ans.PNG

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