Question

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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Answer 1

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Answer 2

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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