Question

A rectangle is inscribed in an equilateral triangle, with one side on a side of the triangle. If the triangle has side of length 2, what is the maximum possible area of the rectangle?

Answer 1

`max. A= sqrt3/2 " units"^2`

Explanation:

let `b and h` be the base and the height of the rectangle `DEFG`, respectively, as shown in the figure.
Given that the equilateral triangle `DeltaABC` has side of length `2, => b+2x=2`,
`=> x=(2-b)/2=1-b/2`
`h/x=tan60, => h=xtan60=sqrt3x`
Area of rectangle `DEFG=|DEFG|=b*h`
`=> |DEFG|=bsqrt3x=sqrt3b(1-b/2)`
`=sqrt3(b-b^2/2)`
`= sqrt3/2(2b-b^2)`
`= -sqrt3/2(b^2-2b)`
`=-sqrt3/2((b-1)^2-1)`
`=sqrt3/2-sqrt3/2(b-1)^2`
`=> max|DEFG|` can be obtained when `b=1`
`=> max. |DEFG|=sqrt3/2 " units"^2`