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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. - Mathematics

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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.

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APPEARS IN

 Selina Solution for Concise Mathematics for Class 10 ICSE (2020 (Latest))
Chapter 18: Tangents and Intersecting Chords
Exercise 18 (B) | Q: 6 | Page no. 283
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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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