Question

10. O and C are respectively the orthocentre and circumcentre of an acute-angled triangle PQR. The
points P and O are joined and produced to meet the side QR at S. If PQS= 60° and
QCR = 130°, then RPS =​

Answer 1

∠RPS = 35°  if O and C are respectively the orthocentre and circumcentre of an acute-angled triangle PQR. The points P and O are joined and produced to meet the side QR at S. If PQS= 60° and QCR = 130°

Step-by-step explanation:

The orthocenter is the point where all three altitudes of the triangle intersect

P and O are joined and produced to meet the side QR at S

=> PS ⊥ QR

=> ∠PSQ = 90°

   ∠PQS = 60°

=> ∠QPS  = 180° - 90° - 60°

=> ∠QPS  = 30°

∠QCR = 30°

C is circumcenter hence center of circle

QR is a chord

=> ∠QCR = 2∠QPR  (angle subtended by chord at center & arc)

=> 130° = 2∠QPR

=> ∠QPR  = 65°

∠QPR = ∠QPS + ∠RPS

=> 65° = 30° + ∠RPS

=> ∠RPS = 35°

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