Question

in a given figure a diameter AB bisects a chord PQ. AQ is parellel to BP. Prove that chord PQ is also a diameter. what is the name given to quadrilateral AQBP

Answer 1

Let AB be the diameter and chord PQ be bisected by AB at O.
⇒op=oq...1
to show pq =diametre
We'll show that O is centre of the circle i.e., OA = OB
We have AQ || BP
⇒ ∠1 = ∠2 (alternate interior angles) ....2
Consider ∆OAQ and ∆OPB 
∠3 = ∠4(vertically opposite angles)
OP = OQ ;from eq 1
∠1 = ∠2 from ...2
∴ ∆OAQ ≅ ∆OPB by asa rule
⇒ OA = OB
⇒ O is the centre of the circle.
∴ PQ is also a diameter of the circle.PROOVED