PQ and RQ are the chords of a circle equidistant from the centre. Prove that the diameter passing through Q bisects ∠PQR and ∠PSR.

Question

PQ and RQ are the chords of a circle equidistant from the centre. Prove that the diameter passing through Q bisects ∠ PQR and ∠ PSR.

Philoid · Accepted Answer

Given: chords PQ and RQ are equidistant from center.

Here consider ΔPQS and ΔRQS

Here,

QS = QS (common)

∠QPS = ∠QRS (right angle)

PQ = QS (chords equidistant from center are equal in length)

∴ By RHS congruency ΔPQS ΔRQS

∴ ∠RQS = ∠SQP and ∠RSQ = ∠QSP (by C.P.C.T)

Therefore we can say that diameter passing through Q bisects and