1. Key Concepts and Formulas

• Rectangle Inscribed in a Circle: The diagonals of the rectangle are diameters of the circle, and the center of the circle is the intersection of the diagonals.
• Distance from a Point to a Line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.
• Distance between two points: The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

2. Step-by-Step Solution

Step 1: Calculate the length of the given side of the rectangle. Let $A = (1, 2)$ and $B = (3, 6)$. We find the length of side $AB$, which we'll call $l$, using the distance formula: $l = \sqrt{(3 - 1)^2 + (6 - 2)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$ This represents the length of one side of the rectangle.

Step 2: Find the slope of the given side and the given diameter. The slope of the side $AB$ is: $m_{AB} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$ The equation of the diameter is $2x - y + 4 = 0$. Rewriting this in slope-intercept form, we get $y = 2x + 4$. Thus, the slope of the diameter is $m_D = 2$.

Step 3: Determine the equation of the line containing side AB. Since we have a point $(1,2)$ on $AB$ and the slope $m_{AB} = 2$, the equation of the line is: $y - 2 = 2(x - 1)$ $y - 2 = 2x - 2$ $2x - y = 0$

Step 4: Find the coordinates of the center of the circle. The center of the circle lies on the diameter $2x - y + 4 = 0$. Let the center be $(h, k)$. Then $2h - k + 4 = 0$. Also, the other side of the rectangle must be perpendicular to the length.

Step 5: Use the distance from the center to side AB to find the length of the other side. The distance from the center $(h, k)$ to the line $2x - y = 0$ is half the length of the other side, say $w$. This distance is: $d = \frac{|2h - k|}{\sqrt{2^2 + (-1)^2}} = \frac{|2h - k|}{\sqrt{5}}$ Since $2h - k + 4 = 0$, we have $2h - k = -4$. Therefore, $d = \frac{|-4|}{\sqrt{5}} = \frac{4}{\sqrt{5}}$ This distance is half the length of the other side $w$, so $w/2 = \frac{4}{\sqrt{5}}$, which means $w = \frac{8}{\sqrt{5}}$.

Step 6: Calculate the area of the rectangle. The area of the rectangle is $A = l \times w$: $A = (2\sqrt{5}) \times \left(\frac{8}{\sqrt{5}}\right) = 2 \times 8 = 16$

3. Common Mistakes & Tips

• Confusing the distance from the center to a side with the length of that side. Remember the distance from the center to a side is half the length of the other side.
• Not using the fact that the center lies on the diameter to simplify calculations.
• Careless errors in applying the distance formulas.

4. Summary

We first found the length of the given side of the rectangle. Then, using the equation of the diameter and the fact that the center of the circle lies on it, we found the perpendicular distance from the center to the side. This distance is half the length of the other side. Finally, we calculated the area of the rectangle by multiplying the lengths of its sides.

The final answer is \boxed{16}.