#### Question

Tangent at P to the circumcircle of triangle PQR is drawn. If the tangent is parallel to side, QR show that ΔPQR is isosceles.

#### Solution

DE is the tangent to the circle at P.

DE ∥ QR (Given)

`∠`EPR = `∠`PRQ (Alternate angles are equal)

`∠`DPQ = `∠`PQR (Alternate angles are equal) ….. (i)

Let `∠`DPQ = x and `∠`EPR = y

Since the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment

∴ `∠`DPQ = `∠`PRQ ……….. (ii) (DE is tangent and PQ is chord)

from (i) and (ii)

`∠` PQR = `∠`PRQ

⇒ PQ = PR

Hence, triangle PQR is an isosceles triangle.

Is there an error in this question or solution?

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Tangent at P to the Circumcircle of Triangle Pqr is Drawn. If the Tangent is Parallel to Side, Qr Show that δPqr is Isosceles. Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.

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