A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (– 8, 5) and (6, 5), then the area of the rectangle (in sq. units) is

Options

A. 84
B. 56
C. 72
D. 98

Explanation

To find the area of the rectangle inscribed in a circle with a diameter along the line \(3y = x + 7\) and with two adjacent vertices at \((-8, 5)\) and \((6, 5)\), we can follow these steps:

### Step 1: Identify the coordinates of the given vertices
The two adjacent vertices of the rectangle are given as:
- Vertex A: \((-8, 5)\)
- Vertex B: \((6, 5)\)

### Step 2: Determine the midpoint of the diagonal AC
Since the rectangle is inscribed in a circle, the diagonal of the rectangle bisects each other at the center of the circle. We need to find the coordinates of the other two vertices, C and D, which will have the same x-coordinates as A and B but different y-coordinates.

Let the coordinates of C and D be:
- Vertex C: \((6, y_1)\)
- Vertex D: \((-8, y_1)\)

### Step 3: Find the midpoint of diagonal AC
The midpoint \(M\) of diagonal \(AC\) can be calculated as follows:
\[ M = \left(\frac{-8 + 6}{2}, \frac{5 + y_1}{2}\right) = \left(-1, \frac{5 + y_1}{2}\right) \]

### Step 4: Find the equation of the line (diameter)
The line along the diameter is given by:
\[ 3y = x + 7 \quad \Rightarrow \quad y = \frac{1}{3}x + \frac{7}{3} \]

### Step 5: Substitute the midpoint into the line equation
Since the midpoint \(M\) lies on the line, we substitute \(x = -1\) into the line equation to find \(y_1\):
\[ y = \frac{1}{3}(-1) + \frac{7}{3} = -\frac{1}{3} + \frac{7}{3} = \frac{6}{3} = 2 \]
Thus, \(y_1 = 2\).

### Step 6: Determine the coordinates of vertices C and D
Now we can find the coordinates of vertices C and D:
- Vertex C: \((6, 2)\)
- Vertex D: \((-8, 2)\)

### Step 7: Calculate the lengths of the sides of the rectangle
The length of side AB (horizontal side) can be calculated as:
\[ AB = |x_2 - x_1| = |6 - (-8)| = |6 + 8| = 14 \]

The length of side AD (vertical side) can be calculated as:
\[ AD = |y_2 - y_1| = |5 - 2| = 3 \]

### Step 8: Calculate the area of the rectangle
The area \(A\) of the rectangle can be calculated as:
\[ A = AB \times AD = 14 \times 3 = 42 \text{ square units} \]

### Final Answer
The area of the rectangle is \(42\) square units.

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