# Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines. NCERT Class X Mathematics - Exemplar Problems Chapter _Circles

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by balaSani416

2015-08-07T18:03:34+05:30

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Given ;-

⇒ A circle wit centre '' O '''.

⇒ And , also O touches Line - l₁ and l₂ .

⇒ l₁ and l₂ lines intersect at  each other

To Prove :-

The centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

Proof ;-

Assume that O touches l₁ and l₂ at M and N , then we get as ,

⇒ OM = ON ( As it is the radius of the circle )

- Therefore ,From the centre  ''O'' of the circle , it  has equal distance from l₁ & l₂.

Now , In Δ OPM & OPN ,

⇒ OM = ON ( Radius of the circle )

⇒Angle - OMP = Angle -ONP ( As , Radius is perpendicular to its tangent )

⇒ OP = OP ( Common sides )

Therefore , Δ OPM = ΔOPN ( S S S congruence rule )

By C.P.C.T ,

⇒ Angle MPO = Angle NPO

So , l bisects angle MPN.

Threfore , O lies on the bisector of the angle between l₁ & l₂ .

Hence , we prove that ⇒ the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.