9. A circle touches the sides of a quadrilateral ABCD at P, Q , R, S respectively. Show that the
angles subtended at the centre by a pair of opposite sides are supplementary.
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9. A circle touches the sides of a quadrilateral ABCD at P, Q , R, S respectively. Show that the
angles subtended at the centre by a pair of opposite sides are supplementary.
answer me
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Answer:
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Step-by-step explanation:
MATHS
A circle touch the sides of a quadrilateral ABCD at P,Q,R,S respectively Show that the angle subtended at the centre by a pair of opposite sides are supplementary.
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ANSWER
A circle the centre O touches the sides AB, BC, CD and DA of a quadrilateral ABCD at the points P,Q,R and S respectively.
To prove: ∠AOB+∠COD=180
o
and, ∠AOD+∠BOC=180
o
Construction: Join OP,OQ,OR and OS
Proof:
Since the two tangents drawn from an external point to a circle subtend equal angles at the centre.
∴∠1=∠2,∠3=∠4,∠5=∠6 and ∠7=∠8
Now, ∠1+∠2+∠3+∠4+∠5+∠6+∠7+∠8=360
⇒2(∠2+∠3+∠6+∠7)=360
and
2(∠1+∠8+∠4+∠5)=360
(∠2+∠3+)+(∠6+∠7)=180
and (∠1+∠8)+(∠4+∠5)=180
[∵∠2+∠3=∠AOB, ∠6+∠7=∠COD, ∠1+∠8=∠AOD \ and \ ∠4+∠5=∠BOC]
⇒∠AOB+∠COD=180
⇒∠AOD+∠BOC=180
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