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# ABC Is An Equilateral Triangle With Side Length 6. If x Is A Point On Side CA, What Is The Minimum Value Of BxÃ¢Â² + XcÃ¢Â²?

Sujoy Das
31/01/2018 0 0

Q. ABC is an equilateral triangle with side length 6. If X is a point on side CA , what is the minimum value of BXÂ² + XCÂ² ?

Answer: Let's D is mid point of CA (BD = height = 3√3)

So, Bx^2 + Cx^2 = (3√3)^2 + Dx^2 + Cx^2 = 27 + Dx^2 + Cx^2

So minima = 27 + 2Dx.Cx (from AM GM)

Now (Dx + Cx) = 3

DX^2 +CX^2 = (9-2DX.CX)

2Dx.Cx = 9-2Dx.Cx

2Dx.Cx = 9/2

Ans: 27+9/2 =  63/2

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