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Acutely Seeking Perimeter (Posted on 2007-09-28) Difficulty: 3 of 5
PQR is an acute angled triangle.

It is known that:
  • S, T and U are points located respectively on the sides QR, RP and PQ.
  • PS is perpendicular to QR.
  • RU is the internal bisector of the angle PRQ.
  • RU intersects PS and ST respectively at the points J and K.
  • PT = TR.
  • UJ=2, JK = 1 and KR = 3
Determine the perimeter of the triangle PQR.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution | Comment 1 of 2
Since K bisects UR and T bisects PR, TK is parallel to PQ. It follows that S bisects QR. So PS bisects angle QPR and the triangle is at least isosceles with apex at P and base QR.

Playing around with Geometer's Sketchpad, the triangle seems it must be uniquely equilateral. That is, only one position for Q relative to PR works, and since an equilateral triangle does work analytically, that is the unique solution.

Half of one side of the triangle comes out to 2*sqrt(3), so the full perimeter is 12*sqrt(3) ~= 20.78460969082653.
  Posted by Charlie on 2007-09-28 11:05:16
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