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Magical Mathematics[Interesting Approach]> maxima-and-minima...

question mark

The maximum area of a reactangle whose sides pass through the angular ponts of a given resctangle of sides a and b is :

ans is 1/2(a+b)2..

plzz explain

Ranjita yadav , 12 Years ago

Grade 11

1 Answers

Jitender Singh

Last Activity: 11 Years ago

Ans:Hello student, please find answer to your question
Let the angular points of the given rectangle to be O (0, 0), A (a, 0), B (a, b) and C (0, b).
Diagonal length:
$\sqrt{a^{2}+b^{2}}$

Since the rectangle passing through angular points of rectangle OABC. Then side length of one side must be equal to diagonal length

Let the
$c + d = \sqrt{a^{2}+b^{2}}$
Let side length of other side be
$e + f$
We have
$e = \sqrt{a^{2}-c^{2}}$
$f = \sqrt{b^{2}-d^{2}}$
Area of rectangle
$A = (c+d)(e+f)$
$A = \sqrt{a^{2}+b^{2}}(\sqrt{a^{2}-c^{2}}+\sqrt{b^{2}-(\sqrt{a^{2}-b^{2}}-c)^{2}})$
$\frac{\partial A}{\partial c} = 0$
$\Rightarrow c = \frac{a^{2}\sqrt{a^{2}+b^{2}}}{2}$
$A_{max} = \frac{(a+b)^{2}}{2}$

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