ABCD is a rectangle with A(-1,2) , B (3,7) and AB : BC = 4 : 3 . If d is the distance from the vertices of the rectangle then [d] is equal to (where [.] denotes the greatest integer function ).

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem, we need to find the distance \( d \) from the vertices of the rectangle ABCD, given the coordinates of points A and B, and the ratio of the lengths of sides AB and BC.

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Given points:
- \( A(-1, 2) \)
- \( B(3, 7) \)

2. **Calculate the Length of AB:**
- The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Plugging in the coordinates of A and B:
\[ AB = \sqrt{(3 - (-1))^2 + (7 - 2)^2} = \sqrt{(3 + 1)^2 + (5)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]

3. **Use the Given Ratio:**
- We know \( \frac{AB}{BC} = \frac{4}{3} \).
- Let \( BC = y \). Then:
\[ \frac{\sqrt{41}}{y} = \frac{4}{3} \implies y = \frac{3 \sqrt{41}}{4} \]

4. **Calculate the Length of AC (Diagonal):**
- In a rectangle, the diagonal can be calculated using the Pythagorean theorem:
\[ AC = \sqrt{AB^2 + BC^2} \]
- Substitute \( AB \) and \( BC \):
\[ AC = \sqrt{(\sqrt{41})^2 + \left(\frac{3 \sqrt{41}}{4}\right)^2} = \sqrt{41 + \frac{9 \cdot 41}{16}} = \sqrt{41 + \frac{369}{16}} = \sqrt{\frac{656 + 369}{16}} = \sqrt{\frac{1025}{16}} = \frac{\sqrt{1025}}{4} \]

5. **Find the Distance \( d \):**
- The distance \( d \) from the vertices of the rectangle to the center can be expressed as:
\[ d = \frac{AC}{2} = \frac{\frac{\sqrt{1025}}{4}}{2} = \frac{\sqrt{1025}}{8} \]

6. **Calculate the Value of \( d \):**
- To find \( \sqrt{1025} \):
\[ \sqrt{1025} \approx 32.0156 \quad \text{(since } 32^2 = 1024\text{)} \]
- Thus:
\[ d \approx \frac{32.0156}{8} \approx 4.00195 \]

7. **Apply the Greatest Integer Function:**
- The greatest integer function \( [d] \) is the largest integer less than or equal to \( d \):
\[ [d] = [4.00195] = 4 \]

### Final Answer:
\[ [d] = 4 \]