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Question Number 56037 by ajfour last updated on 08/Mar/19

Commented by ajfour last updated on 08/Mar/19

Find maximum area of inner triangle if outer one is equilateral.

Findmaximumareaofinnertriangleifouteroneisequilateral.

Commented by mr W last updated on 08/Mar/19

is it not when inner triangle is equal to the outer one?

isitnotwheninnertriangleisequaltotheouterone?

Commented by 121194 last updated on 08/Mar/19

are you sure there not any additional condition there?

areyousuretherenotanyadditionalconditionthere?

Commented by ajfour last updated on 08/Mar/19

a, b, c are not equal.

a,b,carenotequal.

Commented by mr W last updated on 09/Mar/19

thank you for clarifing sir! a,b,c are given and there are many cases, so the question is hard.

thankyouforclarifingsir!a,b,caregivenandtherearemanycases,sothequestionishard.

Answered by ajfour last updated on 11/Mar/19

let side of equilateral △ be s. ⇒ 3s=a+b+c let bottom segments be x and s−x. The right side segments are c−s+x and 2s−x−c. The left side segments are s−b+x, and b−x. For the inner △ area to be maximum, The remaining area has to be minimum. The blue area A=(1/2)sin 60°[x(b−x)+(s−x)(c−s+x) +(2s−c−x)(s−b+x) (dA/dx)=((√3)/4)[(b−x−x)+(−c+s−x+s −x)+(−s+b−x+2s−c−x) ]= 0 ⇒ 3s−6x+2b−2c = 0 (a+b+c)+2b−2c = 6x ⇒ x_0 =((a+3b−c)/6) (d^2 A/dx^2 )= −6 ⇒ A is maximum for x_0 =((a+3b−c)/6) . ⇒ b−x = ((−a+3b+c)/6) Minimum inner △ area A is =((√3)/(36))(a+b+c)^2 −((3(√3))/4)(((9b^2 −(c−a)^2 )/(36))) =((√3)/(36))(a+b+c)^2 − (((√3)[9b^2 −(c−a)^2 ])/(48)) .

letsideofequilateral△bes.⇒3s=a+b+cletbottomsegmentsbexands−x.Therightsidesegmentsarec−s+xand2s−x−c.Theleftsidesegmentsares−b+x,andb−x.Fortheinner△areatobemaximum,Theremainingareahastobeminimum.TheblueareaA=12sin60°[x(b−x)+(s−x)(c−s+x)+(2s−c−x)(s−b+x)dAdx=34[(b−x−x)+(−c+s−x+s−x)+(−s+b−x+2s−c−x)]=0⇒3s−6x+2b−2c=0(a+b+c)+2b−2c=6x⇒x0=a+3b−c6d2Adx2=−6⇒Aismaximumforx0=a+3b−c6.⇒b−x=−a+3b+c6Minimuminner△areaAis=336(a+b+c)2−334(9b2−(c−a)236)=336(a+b+c)2−3[9b2−(c−a)2]48.

Commented by mr W last updated on 10/Mar/19

very nice sir! is this the mininum inner triangle or the maximum?

verynicesir!isthisthemininuminnertriangleorthemaximum?

Commented by ajfour last updated on 11/Mar/19

unfortunately its the minimum area Sir!

unfortunatelyitstheminimumareaSir!

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