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Circle And Intersection (Posted on 2008-01-25) Difficulty: 3 of 5
P, Q and R are three points located on a circle L with diameter 4 and satisfying PQ = QR. Point S is located inside L in such a manner that QR = RS = SQ. The line passing through P and S intersects L at the point T.

Determine the length of ST.

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Solution | Comment 5 of 6 |

QRS is an equilateral triangle,
  QRS = RSQ = SQR = 60
Therefore,
  PQS = PQR - SQR = (OQP + OQR) - SQR = 2x - 60
PQS is an isosceles triangle. Therefore,
  QPS = QSP = 90 - PQS/2 = 120 - x
Therefore,
  RST = 180 - PSQ - QSR = x
PQRT is a cyclic quadrilateral. Therefore,
  QRT = 180 - QPT = 180 - QPS = 60 + x
Therefore,
  SRT = QRT - QRS = x
Therefore,
  SRT = RST implies ST = RT
Therefore,
  ORT = QRT - QRO = 60
The base angle of the isosceles triangle ROT is 60.
Therefore, ROT is an equilateral triangle with
  RT = OT
Hence,
  ST = RT = OT
ST equals the radius of circle L.

  Posted by Bractals on 2008-01-29 17:30:11
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