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What is the least possible length of a line segment that cuts a triangle with sides 3,4,5 in to two geometric figures having equal area?

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    $\begingroup$ What have you tried so far? $\endgroup$
    – Dr. Mathva
    Feb 13, 2019 at 17:02
  • $\begingroup$ Why don't you answer to @Dr. Mthva 5 hours after the question has been asked, immediately after you posted yours ? This site is for exchanges with askers that have really worked on their subject. $\endgroup$
    – Jean Marie
    Feb 13, 2019 at 22:10
  • $\begingroup$ HINT: 3,4,5 is a pythogorean triplet. So it is a right angled triangle $\endgroup$
    – user642405
    Feb 14, 2019 at 6:27

1 Answer 1

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Hints:

  1. Among triangles of equal area with a given angle between two sides, the isosceles triangle has the least possible side opposite to the angle.

  2. Among isosceles triangles of equal area, the triangle with the smallest angle has the least possible side opposite to the angle.

Applying the above considerations and some simple trigonometry you should end up with the answer "2".

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  • $\begingroup$ Did you mean isosceles triangles? $\endgroup$ Feb 14, 2019 at 11:56
  • $\begingroup$ @DanielMathias Of course. Thank you for catching this. $\endgroup$
    – user
    Feb 14, 2019 at 12:00
  • $\begingroup$ Can i get proof of HINT #1. $\endgroup$ Feb 14, 2019 at 13:30
  • $\begingroup$ It is geometrically obvious but you can get it also algebraically minimizing $a^2+b^2-2ab\cos\alpha$ subject to constrain $ab\sin\alpha=2A$. $\endgroup$
    – user
    Feb 14, 2019 at 13:43
  • $\begingroup$ I know this is obvious but i need to proof algebraically. Sorry to say, thing you wrote i didnt get that. Please help me in this. $\endgroup$ Feb 14, 2019 at 14:34

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