O and O 39 are repectively the orthocentre and cicumcentre of an acute angled triangle PQR the point...........

Quantitative Aptitude ( CCS) Coordinate Geometry

O and O' are repectively the orthocentre and cicumcentre of an acute angled triangle PQR. the point P and O are joined and produced to meet the side QR at S. If ∠PQS = 60° and ∠QO'R = 130° then ∠RPS = ?

45°

35°

55°

65°

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Explanation:

∠ PQS = 60°
∠QO'R = 130°
∠ QPR = 1/2 × 130° = 65°
⇒ ∠ QRP = 180° – 60° – 65° = 55°
⇒ ∠PO'Q = 110°
In Δ QO'R
QO' = O'R
⇒ ∠O'QR = ∠O'RQ = 25°
? ∠O'QR + ∠O'RQ = 50°
⇒ ∠PQO' + ∠QPO' = 35°
? ∠PQO' + ∠QPO' = 70°
Similarly, ∠O'PR = 30°
∴ ∠RPS = 35°