In the figure, the tangent from A touches the circle at B. BC is diameter. The line AC cuts the circle at P.
The tangent at P cuts AB at M. Which of the following options are correct?
∠PAM = ∠APM
∠BPM = ∠PBM
M is the midpoint of AB.
Join PB. The figure looks like :
(i)
∠BPM = ∠PBM = ∠PCB = x∘ ...........(angles in the alternate segment are equal)
(ii)
∴ PM = MB ........(base angles of an isosceles △)
(iii)
∠CPB = 90∘ ...........(angle in the semicircle)
(iv)
∴ ∠APB = 90∘ .........(linear pair)
(v)
∠APM = 90∘ - x∘
(vi)
∠PMA = 2x∘ ...........(exterior angle)
(vii)
∠PAM = 90∘ - x∘ .........(Sum of angles of a △)
(viii) ∴ ∠APM = ∠PAM = 90∘ - x∘ ..........from (v) and (vii)
(ix) PM = AM .........(Base angles of an isosceles triangle)