A (-3,4) , B (5,4) , C and D form a rectangle `X - 4y+7=0` is a diameter of the circumcirle of the rectangle ABCD, the area of `DeltaABC` is

Options

A. 8
B. 16
C. 32
D. 64

Answer

To solve the problem, we need to find the area of triangle ABC formed by points A, B, and C, given that A(-3, 4) and B(5, 4) are two vertices of a rectangle ABCD, and the line \(X - 4y + 7 = 0\) is the diameter of the circumcircle of the rectangle.

### Step-by-Step Solution:

1. **Identify Points A and B**:
- Point A is given as (-3, 4).
- Point B is given as (5, 4).

2. **Determine the Midpoint of AB**:
- The midpoint \(E\) of segment \(AB\) can be calculated using the midpoint formula:
\[ E = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{-3 + 5}{2}, \frac{4 + 4}{2}\right) = \left(1, 4\right) \]

3. **Find the Equation of the Perpendicular Bisector of AB**:
- Since AB is horizontal (y = 4), the perpendicular bisector will be a vertical line passing through E.
- Therefore, the equation of the perpendicular bisector is \(x = 1\).

4. **Find the Intersection of the Perpendicular Bisector with the Diameter**:
- The diameter of the circumcircle is given by the line equation \(x - 4y + 7 = 0\).
- Substitute \(x = 1\) into the line equation:
\[ 1 - 4y + 7 = 0 \implies 4y = 8 \implies y = 2 \]
- Thus, the center of the circumcircle (and also the midpoint of the rectangle) is at point \(C(1, 2)\).

5. **Identify the Coordinates of Point D**:
- Since the rectangle is symmetric about the midpoint, point D will have the same x-coordinate as point C and the same y-coordinate as point A.
- Thus, point D will be at \(D(1, 4)\).

6. **Calculate the Area of Triangle ABC**:
- The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, the base \(AB\) is the distance between points A and B:
\[ AB = |x_2 - x_1| = |5 - (-3)| = 8 \]
- The height is the vertical distance from point C to line AB (which is at y = 4):
\[ \text{Height} = |y_C - y_{AB}| = |2 - 4| = 2 \]
- Now, substituting these values into the area formula:
\[ \text{Area} = \frac{1}{2} \times 8 \times 2 = 8 \]

### Final Answer:
The area of triangle ABC is \(8\).