In the given figure, P is the mid-point of arc APB and M is the mid-point of chord AB of a circle with centre O. Prove that:

(i) PM ⟂ AB

(ii) PM produced will pass through the centre O

(iii) PM produced will bisect the major arc AB.

(i) Given,

P is the mid-point of arc APB.

Thus, arc AP = arc PB.

We know that,

Equal arcs subtends equal chords.

Thus,

chord AP = chord PB.

In triangle APB,

AP = PB

Thus, APB is an isosceles triangle.

In an isosceles triangle the median drawn from common vertex to opposite side is also perpendicular.

Thus, PM ⊥ AB.

Hence, proved that PM ⟂ AB.

(ii) The perpendicular bisector of any chord of a circle passes through the centre.

Hence, proved that PM produced will pass through the centre O.

(iii) A line through the centre and the mid-point of a chord will also bisect the corresponding arc.

Since, M is the mid-point of chord AB and PM passes through center O.

Hence, proved that PM produced will bisect the major arc AB.