Between the various triangles on same base and between the same parallel lines which one would have the shortest perimeter... isosceles, equilateral or other? how can we prove it?

Sreejato, I would disagree with you. The main issue lies in understanding what Rahul meant in his problem, which could be stated clearer.

My interpretation of the problem is as follows: We are given 2 points \(A\) and \(B\) and a line \( \ell\) which is parallel to \( AB\). Consider all triangles \(ABC\) with \(C\) on \(\ell\). Which triangle has minimal perimeter?

@Sreejato Bhattacharya
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I don't really think this is correct. I believe you meant \(A=(0,0), B=(c,0), C=(x,h). \) Perimeter\(=c+\sqrt{x^2+h^2}+\sqrt{(c-x)^2+h^2}\). Ignore the \(c\) since it is a constant and by AM-GM the minimum is obtained when \(\sqrt{x^2+h^2}=\sqrt{(c-x)^2+h^2} \Rightarrow x=c-x \Rightarrow x=c/2\), which is an isosceles triangle. Anyway I prefer Gabriel's solution.

@Sreejato Bhattacharya
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Yes. There is a much more direct approach to this problem, where the triangle inequality hides all the algebraic expressions that you use.

Using Calvin's intepretation, we have: Villages A, and B lie on a straight line. There is a river parallel to the line containing A and B. What is the fastest way to run from A to the river and back to B? The answer is simple: reflect B across the river to obtain B'. The distance from the river to B and distance to B', for any point on the river, is the same. So the best point (C) on the river is the intersection of AB' and the river, which makes triangle ABC iscoceles.

Between the various triangles on same base and between the same parallel lines which one would have the shortest perimeter... isosceles or other? how to prove it?
think of the case when base length is less than the perpendicular distance between parallel lines?

@Sreejato Bhattacharya
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when perpendicular distance between two parallel lines is greater then the height of equilateral triangle formed by using base length as length of sides.......

@Rahul Vernwal
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Please pardon me if I don't understand this properly but the perpendicular distance between two parallel lines when perpendicular distance between the two parallel lines is the height of equilateral triangle formed by using base length as length of sides.

Let the length of the base be taken as x. If the perpendicular distance is x√3 / 2 , then, the perimeter of equilateral triangle is 3x. The perimeter of right angled triangle is 3.025x . In this case Equilateral triangle and isosceles triangle is the same. So between 3x and 3.025x, 3x is smaller and equilateral triangle has shorter perimeter.

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TopNewestSreejato, I would disagree with you. The main issue lies in understanding what Rahul meant in his problem, which could be stated clearer.

My interpretation of the problem is as follows: We are given 2 points \(A\) and \(B\) and a line \( \ell\) which is parallel to \( AB\). Consider all triangles \(ABC\) with \(C\) on \(\ell\). Which triangle has minimal perimeter?

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Sir then is the length \( AB \) and the perpendicular distance from \( \ell \) constant?

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Comment deleted Apr 04, 2013

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Hint: Reflect \(B\) across line \(\ell\).

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Using Calvin's intepretation, we have: Villages A, and B lie on a straight line. There is a river parallel to the line containing A and B. What is the fastest way to run from A to the river and back to B? The answer is simple: reflect B across the river to obtain B'. The distance from the river to B and distance to B', for any point on the river, is the same. So the best point (C) on the river is the intersection of AB' and the river, which makes triangle ABC iscoceles.

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Between the various triangles on same base and between the same parallel lines which one would have the shortest perimeter... isosceles or other? how to prove it? think of the case when base length is less than the perpendicular distance between parallel lines?

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Comment deleted Apr 04, 2013

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and what if the triangle cannot be equilateral? i mean if it cannot be formed that way

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Please clarify how it can be formed.

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I'm going to manually edit the votes such that your post doesn't appear at the top.

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Fantastic solution!!. I also got the same answer but after using triogonometry, but you got it so effortlessly

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right sreejato

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it depends on lengths of rest two sides

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Let the length of the base be taken as x. If the perpendicular distance is x√3 / 2 , then, the perimeter of equilateral triangle is 3x. The perimeter of right angled triangle is 3.025x . In this case Equilateral triangle and isosceles triangle is the same. So between 3x and 3.025x, 3x is smaller and equilateral triangle has shorter perimeter.

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