P is the mid-point of an arc QPR of a circle.show the tangent at P is parallel to the chord QR
Let DPE be the tangent at P and QR be the chord of the circle.
To prove: DPE || QR
Given, P is the mid-point of an arc QPR of the circle.
∴ arc QP = arc PR
⇒ QP = PR [If two arcs of a circle are equal then their corresponding chords are also equal.]
Now, in ΔPQR
PQ = PR (Proved above)
∴ ∠PQR = ∠PRQ ...(1) (Equal sides have equal angles opposite to them)
We know that, If a line touches a circle and from the point of contact, a chord is drawn, then the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternative segment.
Now, DPE is the tangent and QR is the chord.
∴ ∠DPQ = ∠PRQ ...(2)
From (1) and (2), we have
∠PQR = ∠DPQ
∴ DPE || QR (If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel)
Thus, the tangent at P is parallel to the chord QR of the circle.