a tangent PT is drawn parallel to a chord AB of a circle . Prove that APB is an Isocles Triangle :
Answers
GIVEN:-
A tangent PT is drawn parallel to a chord AB of a circle
TO PROVE:-
APB is an isosceles triangle
CONSTRUCTION:-
Join OP and OA ,where O is the centre of the circle
PROOF:-
∠OPT=90⁰⠀⠀⠀⠀⠀⠀[Angle between radius and tangent]
➠∠APT+∠APO=90⁰
➠∠APO=90⁰-∠APT......(1)
In △OAP
∵ ⠀⠀⠀⠀⠀OA=OP
⠀⠀⠀⠀⠀⠀⠀⠀[radii of same circle]
∴⠀⠀ ⠀⠀⠀∠APO=∠PAO
⠀⠀⠀⠀⠀⠀⠀⠀[Angles opposite to equal side of a triangle are equal]
➠ ∠AOP=180⁰-(∠APO+∠PAO)
⠀⠀⠀⠀⠀⠀=180⁰-{(90⁰-∠APT)+(90⁰-∠APT)}⠀⠀⠀⠀[from (1)]
⠀⠀⠀⠀⠀⠀=2∠APT
➠2∠ABP=2∠APT
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀[property of circle]
➠∠ABP=∠APT.......(2)
∵PT || AB
∴∠APT=∠PAB........(3)
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀[alt.Int.∠s]
From (2) and (3) ,
∠ABP=∠PAB
∴ PA=PB
⠀⠀⠀⠀⠀⠀[Sides opposite to equal angles of a triangle are equal]
∴△APB is an isosceles triangle
ADDITIONAL INFORMATION
1.No tangent can be drawn to a circle from a point lying inside the circle
2.There is one and only one tangent to a circle passing through a point lying on circle
3.Exactly two tangent can be drawn to a circle from a point outside the circle.
4.The length of tangents drawn from an external point to a circle are equal
