Question

show that it the perimeter of a triangle is constant its area is maximum when the is triangle equilateral​

Answer 1

Answer:

Answer

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amitnrw

Given : perimeter of the triangle is constant

To Find : its area is maximum When it is equilateral triangle

Solution:

a + b + c = 2s ( 2s is constant)

Area A = √s(s - a)(s - b)(s - c)

c = 2s - a - b

=> A = √s(s - a)(s - b)(s - (2s-a - b)))

=> A = √s(s - a)(s - b)(a + b - s))

A is max then A² is also max as A is +ve

Z= A² = s(s - a)(s - b)(a + b - s)

∂Z/∂a = s(-1)(s - b)(a + b - s) + s(s - a)(s - b)(1)

= - s(s - b)(a + b - s) + s(s - a)(s - b)

= s(s - b)( - a - b + s + s - a)

= s( s - b)( 2s - 2a - b)

s = 0 not possible

s = b

2s - 2a - b = 0

Similarly

∂Z/∂b = s( s - a)( 2s - 2b - a)

s = a

2s - 2b - a = 0

if we consider s = b and s= a then c = 0 which is not possible

2s - 2a - b = 0

2s - 2b - a = 0

solving a = b = 2s/3

=> c = 2s/3

a = b = c = 2s/3

∂²Z/∂a² = s( s - b)( - 2) < 0 and ∂²Z/∂b² = s( s - a)( - 2) < 0

Hence Maximum Z when a = b = c = 2s/3

Hence Area is maximum when a = b = c

Hence triangle is Equilateral triangle

QED

if the perimeter of the triangle is constant its area is maximum. When it is equilateral triangle