prove that the centre of a circle touching two intersecting lines lies on the angle bisetor of the lines
Answers
Answer:
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.
Answer:
Step-by-step explanation:
i) Let the two lines l₁ and l₂ intersect at point P. And the circle touches the two lines at A and B respectively. Join center O of the circle with A and B respectively. Also join OP.
ii) At the point of contact radius and tangent are perpendicular.
So, <OAP = <OBP = 90 deg
OP = OP [Common]
OA = OB [Radii of the same circle]
Hence ΔOAP ≅ ΔOBP [RHS Congruence axiom]
So <OPA = <OPB [Corresponding parts of congruence triangles are equal]
Hence OP is the bisector of angle APB.
Thus it is proved that O lies on the bisector of the angle formed by the intersecting lines.