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A rectangle is inscribed in an isosceles triangle with base 1 unit and height h units. what should be the height of the triangle, so that it has minimum area?

A rectangle is inscribed in an isosceles triangle with base 1 unit and height h units. what should be the height of the triangle,
so that it has minimum area?

Grade:11

1 Answers

bharat bajaj IIT Delhi
askIITians Faculty 122 Points
7 years ago

There is some correction in question. The rectangle cant have any minimum area. We will find out the maximum area.
Let the triangle be ABC with BC on the x axis and point A on y axis.
A(0,h) B(-1/2,0) C(1/2,0)
Say the rectangle be EFGH with EF side on side BC.
E(-x,0) F(x,0) G(x,y) H(-x,y)
Now the points A G and C lie on same line. Hence, from slope
-2h = y/(x-1/2)
y = h - 2hx
Now the area of rectangle A = 2xy = 2xh(1-2x)
Differentiate it : 2h( 1 - 4x) = 0
Hence, x = 1/4, y = h/2
These are the conditions for the maximum area by rectangle. The question is incomplete. Hence, these are the final answers.

Thanks
Bharat Bajaj
IIT Delhi
askiitians faculty

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