 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Circle And Intersection (Posted on 2008-01-25) 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) Comments: ( Back to comment list | You must be logged in to post comments.) 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 Please log in:

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