Question

can you solve..............

Answer 1

Answer:

As you have already marked, the angle ∠BPO = 60° since AB and OP are perpendicular and the sum of the angles in a triangle is 180°.

The angle subtended at the centre of a circle is twice the angle on the circumference so ∠AOP = 2∠ABP = 60°.  Since ∠AOP = ∠BPO, the lines BP and AO are parallel.

The radius perpendicular to a chord bisects that chord, so P is in the middle of the chord AB and Q is in the middle of the chord AC.  Consequently, the triangles ABP and ACQ are isosceles.  Also, since AB and AC are equal, the triangles ABP and ACQ are congruent.  Therefore ∠ACQ = ∠ABP = 30°.

As before, we now have ∠CQO = 60° and ∠AOQ = 2∠ACQ = 60°.  This tells us that CQ is parallel to AO.

Since they're both parallel to AO, the lines BP and CQ are parallel to each other.