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

**Proof:**

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.

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