Question

In a figure, PT is a tangent to the circle and PBA is the secant .TM is the bisector pf angle ATB meeting AB at M .Prove that PT = PM

Answer 1

As angle PTA = angle PTB ( tangent -angle theorem) (1)
And Angle BTM = angle MTA  (given) (2)
And angle PMT = angle PBT + angle MTB
So angle PMT = angle PTA + angle MTA (using (1) and (2) )
And angle PTA + angle MTA = PTM
So angle PMT = angle PTM ( hence proved)

In triangle PMT
We have angle PMT = angle PTM
So PM = PT (isosceles  triangle)