Question

two tangents TP and TQ are drawn to a circle with Centre O from an external point T prove that angle Angle PTQ = 2 Angle OPQ

Answer 1

Aftr angl OPT=90degre
=>angle OPQ+angleTPQ=90 degre
=>ANGLE OPQ= 90degre-anglTPQ
now frm 1 and 2
half angl PTQ=anglOPQ
=>angle PTQ=2angleOPQ
THATS IT I HOPE U CAN Undrstnd :-)

Answer 2

Given:- A circle with centre O and TP and TQ are the tangents on it from a point outside T.
To Prove :- angle PTQ = 2 angle OPQ.
Proof:- Let Angle PTQ = X°
We know that the tangents to a circle from an external point are equal.So PA = PB.
The angles opposite to the equal sides of a triangle are also equal,
Therefore,
PA = PB
angle TQP =angle OPQ
Also,
The sum of the angles of a triangle is 180°
Therefore,
Angle PTQ + angle TPQ + angle TQP = 180°
X° + 2angleTPQ= 180°
angle TPQ = 1/2(180°-X°)= (90°-1/2X°)
TP is a tangent and OT is the radius of a circle.
Therefore,
angle OPQ + angle TPQ = 90°
Angle OPQ = (90-1/2×X°) = 90°
angle OPQ = 1/2X° = 1/2×PTQ [ PTQ = X°]
Therefore,
PTQ = 2angle OPQ.......(PROVED)