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shortest perimeter

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?

Note by Rahul Vernwal
3 years, 9 months ago

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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? Calvin Lin Staff · 3 years, 9 months ago

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@Calvin Lin Sir then is the length \( AB \) and the perpendicular distance from \( \ell \) constant? Sreejato Bhattacharya · 3 years, 9 months ago

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Comment deleted Apr 04, 2013

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@Sreejato Bhattacharya 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. Yong See Foo · 3 years, 9 months ago

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

Hint: Reflect \(B\) across line \(\ell\). Calvin Lin Staff · 3 years, 9 months ago

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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. Gabriel Wong · 3 years, 9 months ago

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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? Rahul Vernwal · 3 years, 9 months ago

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Comment deleted Apr 04, 2013

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@Sreejato Bhattacharya and what if the triangle cannot be equilateral? i mean if it cannot be formed that way Rahul Vernwal · 3 years, 9 months ago

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@Rahul Vernwal Please clarify how it can be formed. Sreejato Bhattacharya · 3 years, 9 months ago

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@Sreejato Bhattacharya 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 · 3 years, 9 months ago

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@Rahul Vernwal According to your specified constraints, an isosceles triangle is the best choice for minimizing the perimeter. Adam Silvernail · 3 years, 9 months ago

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@Rahul Vernwal what does it mean?? type the full question Superman Son · 3 years, 9 months ago

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@Superman Son @ rahul Superman Son · 3 years, 9 months ago

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@Rahul Vernwal 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. Sreejato Bhattacharya · 3 years, 9 months ago

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@Sreejato Bhattacharya equilateral<isoceles<abstract<right David Christopher · 3 years, 9 months ago

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@David Christopher yep Superman Son · 3 years, 9 months ago

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@Sreejato Bhattacharya I'm going to manually edit the votes such that your post doesn't appear at the top. Calvin Lin Staff · 3 years, 9 months ago

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@Sreejato Bhattacharya Fantastic solution!!. I also got the same answer but after using triogonometry, but you got it so effortlessly Pratik Singhal · 3 years, 9 months ago

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@Sreejato Bhattacharya right sreejato Superman Son · 3 years, 9 months ago

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it depends on lengths of rest two sides Anubhav Singh · 3 years, 9 months ago

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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. Vasavi GS · 3 years, 9 months ago

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