ABCD is a square. EFGH is a rectangle inscribed within the square with its sides parallel to the diagonal AC of the square. The perimeter of the rectangle EFGH is 16. The side of the square is

Correct option is B

Given :
The perimeter of rectangle EFGH is 16
Forumla Used : 
$$\text{Diagonal} = \sqrt{(\text{Length})^2 + (\text{Width})^2}$$​
Perimeter=2×(Length+Width)
Solution : 
Let the side of the square be  Then the diagonal AC has length $$\sqrt{2}$$ (by the Pythagorean theorem).
The perimeter of EFGH is given as 16, so
2×(Length+Width)=16⟹Length+Width=8
Let the sides of the rectangle parallel and perpendicular to the diagonal be x and , respectively. Thus:
x+y=8
From the equation x+y=8, we substitute y=8−x into the diagonal equation:
​$$\sqrt{x^2 + (8-x)^2}$$ = s$$\sqrt{2}$$
Solving the above equation
​$$x^2 - 8x + 32 = s^2$$
$$s = \sqrt{x^2 - 8x + 32}$$
The possible solutions for  are
​​​$$x = 4 - 4\sqrt{3} \, \text{or} \, x = 4 + 4\sqrt{3}$$
Since must be a positive length and less than 8 (as x+y=8), the valid solution is:
​​$$x = 4 - 4\sqrt{3}$$
Using this, we will calculate ,the side of the square
The side length of the square is approximately s=8.
Thus, the side of the square ABCD is 8 units
Thus the correct answer is option (B) 8
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