PQ is a diameter of a circle and PR is the chord such that angle RPQ =30 DEGREE the tangent - Maths - Circles

Question

PQ is a diameter of a circle and PR is the chord such that
angle RPQ =30 DEGREE the tangent at R intersects PQ produced at S
Prove that QR =QS Figure of the question is given below

Shruti Tyagi · Accepted Answer

Dear student,
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 Given : A circle with PQ as diameter having chord PR
  ∠RPQ = 30°  Tangent at R meets PQ produced at S
To prove : QR=QS Proof : In Δ ROP,  OR = OP         (radii of same circle)  ⇒ ∠1 = ∠RPQ  (angles opposite to equal sides are equal)  ⇒∠1 = 30°  By angle sum property of Δ,We have, ∠2 = 180° – (30°+30°)                      ⇒∠2 = 180°-60°                       ⇒∠2 = 120°  Now, ∠2 + ∠3 = 180°  (linear pair)  ⇒ 120° + ∠3 = 180°  ⇒ ∠3= 60°PQ is diameter of the circleWe know that angle in a semi circle is 90°
  ⇒ ∠PRQ = 90°  ⇒ ∠1 + ∠4 = 90°  ⇒ 30° + ∠4 = 90°  ⇒ ∠4 = 60°Consider OR is radius and SR is tangent to circle at R
  We have OR⊥SR ⇒ ∠ORS=90°  ⇒ ∠4 + ∠5 (=∠SRQ) = 90°  ⇒ 60° + ∠5 = 90°  ⇒ ∠5 = 30°In ΔORS, by angle sum property of Δ  ∠5 + ∠ORS + ∠6 = 180°  ⇒ 60° + 90° + ∠6 = 180°  ⇒ ∠6 + 150° = 180°  ⇒ ∠6 = 30°  In ΔQRS , ∠5 = ∠6        --(each  30°)  ⇒ QS=QR     -- (sides opposite to equal angles are equal)  Hence Proved

Regards

PQ is a diameter of a circle and PR is the chord such that angle RPQ =30 DEGREE the tangent at R intersects PQ produced at S . Prove that QR =QS. Figure of the question is given below.