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Step 1: Identify the given points and understand the configuration
We have two adjacent vertices of the rectangle:
A = (-8, 5) and B = (6, 5).
These points lie on a line that is parallel to the x-axis, since their y-coordinates are the same.
Step 2: Compute the length of side AB
Since both points have the same y-coordinate (5), the distance between A and B is simply the difference in their x-coordinates:
$AB = |x_B - x_A| = |6 - (-8)| = 6 + 8 = 14.$
Step 3: Find the midpoint P of segment AB
The midpoint P of AB is given by:
$P = \left( \dfrac{x_A + x_B}{2},\ \dfrac{y_A + y_B}{2} \right).$
Substituting A = (-8, 5) and B = (6, 5), we get:
$P = \left( \dfrac{-8 + 6}{2},\ \dfrac{5 + 5}{2} \right) = \left( -1,\ 5 \right).$
Step 4: Recognize that OP is perpendicular to AB
OP is drawn from the center O of the circle perpendicular to AB. Since AB is parallel to the x-axis, OP must be parallel to the y-axis. This implies the x-coordinate of O is the same as that of P, which is -1.
Step 5: Determine the center O using the given line
We are told that the diameter of the circle (and hence its center O) lies on the line $3y = x + 7.$
Knowing the x-coordinate of O is -1, substitute $x = -1$ into the line equation:
$3y = (-1) + 7 \implies 3y = 6 \implies y = 2.$
Therefore, the center of the circle is
$O = (-1,\ 2).$
Step 6: Compute the distance OP
The length OP can be found using the coordinates of O and P:
$OP = \sqrt{ (x_O - x_P)^2 + (y_O - y_P)^2 }
= \sqrt{ (-1 - (-1))^2 + (2 - 5)^2 }
= \sqrt{ 0^2 + (-3)^2 }
= 3.$
Step 7: Relate OP to the other side of the rectangle
Since O is the center of the circle, and AB is one side of the rectangle, the other side (let us call it BC) will be twice OP (because OP is from the center to the midpoint of the side of the rectangle inscribed in the circle). Hence,
$BC = 2 \times OP = 2 \times 3 = 6.$
Step 8: Calculate the area of the rectangle
The sides of the rectangle are $AB = 14$ and $BC = 6.$ Therefore, its area is
$ \text{Area} = AB \times BC = 14 \times 6 = 84.$
Correct Answer: 84
