In rhombus ABCD, points E,F,G, and H are the midpoints of lines AB,BC,CD, and DA respectively.
Quadrilateral EFGH has area 14 and perimeter 16.
Find the side length for rhombus ABCD.
\(\text{Let $EF$ = x } \\ \text{Let $EH$ = y } \\ \text{The side length for rhombus $=AB=BC=CD=DA=HF$ = s } \)
\(\begin{array}{|lrcll|} \hline 1. & xy&=&14 \\ & y&=& \dfrac{14}{x} \\\\ 2. & 2(x+y)&=& 16 \\ & \mathbf{x+y} & \mathbf{=} & \mathbf{8} \quad & | \quad y= \dfrac{14}{x}\\ & x+\dfrac{14}{x} &=& 8 \quad & | \quad \cdot x \\ & x^2+ 14 &=& 8x \\ & x^2-8x+ 14 &=& 0 \\ & x &=& \dfrac{8\pm \sqrt{64-4\cdot 14} }{2} \\ & x &=& \dfrac{8\pm \sqrt{4(16-14)} }{2} \\ & x &=& \dfrac{8\pm 2\sqrt{ 16-14 } }{2} \\ & x &=& \dfrac{8\pm 2\sqrt{ 2 } }{2} \\ & x & = & 4\pm \sqrt{ 2 } \\ & \mathbf{x} & \mathbf{=} & \mathbf{4- \sqrt{ 2 }} \\\\ & \mathbf{x+y} & \mathbf{=} & \mathbf{8} \quad & | \quad x=4-\sqrt{ 2 } \\ & 4-\sqrt{ 2 }+y &=& 8 \\ & y &=& 8-(4-\sqrt{ 2 }) \\ & y &=& 8-4+\sqrt{ 2 }) \\ & \mathbf{y} & \mathbf{=} & \mathbf{4+ \sqrt{ 2 }} \\ \hline \end{array} \)
\(\mathbf{s =\ ?}\)
\(\begin{array}{|rcll|} \hline s^2 &=& x^2+y^2 \quad & | \quad \mathbf{x= 4- \sqrt{ 2 }},~ \mathbf{y= 4+ \sqrt{ 2 }} \\ s^2 &=& (4-\sqrt{2})^2+(4+\sqrt{2})^2 \\ s^2 &=& 16-8\sqrt{2} + 2 + 16 +8\sqrt{2} + 2 \\ s^2 &=& 16 + 2 + 16 + 2 \\ s^2 &=& 36 \\ \mathbf{s} & \mathbf{=} & \mathbf{6} \\ \hline \end{array}\)
The side length for rhombus ABCD is 6
