Key Concepts and Formulas

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Midpoint Formula: The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
• Properties of a 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 (and thus the center of the rectangle).

Step-by-Step Solution

Step 1: Calculate the length of side AB and find its midpoint.

We are given the coordinates of vertices A and B as $A(-8, 5)$ and $B(6, 5)$. We will first find the length of side AB and then its midpoint.

• Calculate the length of AB: $AB = \sqrt{(6 - (-8))^2 + (5 - 5)^2} = \sqrt{(14)^2 + 0^2} = \sqrt{196} = 14$ Therefore, the length of side AB is 14 units.

• Find the midpoint of AB (let's call it P): $P = \left(\frac{-8 + 6}{2}, \frac{5 + 5}{2}\right) = \left(\frac{-2}{2}, \frac{10}{2}\right) = (-1, 5)$ The midpoint P of side AB is $(-1, 5)$.

Step 2: Determine the coordinates of the center of the circle (O).

Since the rectangle is inscribed in a circle, the center of the circle is also the center of the rectangle. We know that the diameter of the circle lies on the line $3y = x + 7$. Also, the perpendicular bisector of AB passes through the center of the circle.

• Find the equation of the perpendicular bisector of AB: Since AB is a horizontal line segment (y = 5), its perpendicular bisector is a vertical line passing through the midpoint P(-1, 5). Thus, the equation of the perpendicular bisector is $x = -1$.

• Find the intersection of the perpendicular bisector and the diameter line: The center of the circle (O) lies on both the line $x = -1$ and the line $3y = x + 7$. Substituting $x = -1$ into the equation of the diameter line, we get: $3y = -1 + 7$ $3y = 6$ $y = 2$ Therefore, the coordinates of the center of the circle are $O(-1, 2)$.

Step 3: Calculate the length of the other side of the rectangle (BC).

Let the other side of the rectangle be BC. Since O is the center of the rectangle, the distance from O to the midpoint P of AB is half the length of the side BC.

• Calculate the distance OP: $OP = \sqrt{(-1 - (-1))^2 + (5 - 2)^2} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3$ So, OP = 3.

• Determine the length of BC: Since OP is half the length of BC, we have $BC = 2 \times OP = 2 \times 3 = 6$.

Step 4: Calculate the area of the rectangle.

The area of the rectangle is given by the product of the lengths of its adjacent sides, AB and BC.

• Calculate the area: Area = $AB \times BC = 14 \times 6 = 84$

The area of the rectangle is 84 sq. units.

Step 5: Find the coordinates of the other two vertices. We have A(-8, 5), B(6, 5), and O(-1, 2). We know BC = 6 and AD = 6. Also, BC is vertical. Let C have coordinates (6, y). The midpoint of BC is M, and OM is perpendicular to BC. Since BC = 6, M must be at a distance of 3 from B(6, 5) in either direction. So M = (6, 5+3) = (6, 8) or M = (6, 5-3) = (6, 2).

Since O(-1, 2) is the center of the rectangle, the midpoint of AC is O.

Let C be (x, y). Then $\frac{x - 8}{2} = -1$ and $\frac{y+5}{2} = 2$. $x - 8 = -2 => x = 6$. $y+5 = 4 => y = -1$. So C = (6, -1). Similarly, D = (-8, -1).

Then BC = $\sqrt{(6-6)^2 + (5 - (-1))^2} = 6$. CD = $\sqrt{(6 - (-8))^2 + (-1 - (-1))^2} = 14$. DA = $\sqrt{(-8 - (-8))^2 + (-1 - 5)^2} = 6$. Area = 14*6 = 84.

Common Mistakes & Tips

• Be careful with signs when using the distance and midpoint formulas.
• Remember that the center of the circle is also the center of the inscribed rectangle.
• The perpendicular bisector of a chord always passes through the center of the circle.

Summary

We found the length of one side of the rectangle using the given vertices. Then, we located the center of the circle (and rectangle) by finding the intersection of the perpendicular bisector of that side and the line containing the diameter. Using the distance from the center to the midpoint of the first side, we calculated the length of the adjacent side and, finally, the area of the rectangle.

The final answer is \boxed{84}, which corresponds to option (B).