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Take Unit Length, Get PR (Posted on 2007-06-17) Difficulty: 3 of 5
The perpendicular from vertex P of the triangle PQR meets QR at the point S. A point T is located on PR such that QT=TR=RS=1.

What is the length of PR, given that Angle QPR=90o?

  Submitted by K Sengupta    
Rating: 3.0000 (1 votes)
Solution: (Hide)
Let the lengths of the three sides QR, PR and PQ be respectively denoted by a, b and c.

Since PS is perpendicular to QR, it follows that the triangles PQR and PSR are similar, giving:

PR/ QR = RS/ PR
Or, RS = b^2/a = 1, giving: a = b^2..(i)

Also from Triangle QPT, w observe that:
QT^2 = PQ^2 + PT^2 = PQ^2 + (PR TR)^2
Or, 1 = c^2 + (b-1)^2

But, we know that: c^2 = a^2 b^2

Therefore:
1 = c^2 + (b-1)^2
= a^2 b^2 +(b-1)^2
= a^2 2b + 1
= b^4 2b + 1

This yields, b^4 2b = 0, so that: b = 3√(2)

Consequently, the required length of PR is 3√(2)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolution (other way)Praneeth Yalavarthi2007-07-10 07:13:14
SolutionSolutionBractals2007-06-17 13:26:44
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