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 step by step, we will follow the given information about the rectangle ABCD, where A(-1, 2) and B(3, 7) are two vertices. We also know that the ratio of the lengths of sides AB to BC is 4:3. We need to find the distance from the center of the rectangle to its vertices and then apply the greatest integer function.

### Step 1: Calculate the length of AB
Using the distance formula, we can find the length of AB. The distance formula between two points (x1, y1) and (x2, y2) is given by:

\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of points A and B:

\[ AB = \sqrt{(3 - (-1))^2 + (7 - 2)^2} = \sqrt{(3 + 1)^2 + (5)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]

### Step 2: Use the ratio of AB to BC
Given that the ratio of AB to BC is 4:3, we can express BC in terms of AB:

\[ \frac{AB}{BC} = \frac{4}{3} \implies BC = \frac{3}{4} AB \]

Substituting the value of AB:

\[ BC = \frac{3}{4} \cdot \sqrt{41} \]

### Step 3: Calculate the length of AC using Pythagorean theorem
In rectangle ABCD, we can use the Pythagorean theorem to find the diagonal AC:

\[ AC^2 = AB^2 + BC^2 \]

Substituting the values:

\[ AC^2 = (\sqrt{41})^2 + \left(\frac{3}{4} \sqrt{41}\right)^2 \]

Calculating each term:

\[ AC^2 = 41 + \frac{9}{16} \cdot 41 = 41 + \frac{369}{16} = \frac{656 + 369}{16} = \frac{1025}{16} \]

Thus,

\[ AC = \sqrt{\frac{1025}{16}} = \frac{\sqrt{1025}}{4} \]

### Step 4: Find the distance from the center to the vertices
The distance from the center (O) of the rectangle to any vertex (D) is half of the diagonal AC:

\[ d = \frac{1}{2} AC = \frac{1}{2} \cdot \frac{\sqrt{1025}}{4} = \frac{\sqrt{1025}}{8} \]

### Step 5: Calculate the greatest integer function
To find [d], we need to evaluate \(\frac{\sqrt{1025}}{8}\):

First, we approximate \(\sqrt{1025}\):

\[ \sqrt{1025} \approx 32.0156 \implies d \approx \frac{32.0156}{8} \approx 4.00195 \]

Thus, the greatest integer function [d] is:

\[ [d] = 4 \]

### Final Answer
The value of [d] is **4**.