The quadrilateral ABCD formed by the points A (0, 0), B (3, 4), C (7, 7) and D (4, 3) is a

Options

A. rectangle
B. square
C. rhombus
D. parallelogram

Answer

To determine the type of quadrilateral formed by the points A(0, 0), B(3, 4), C(7, 7), and D(4, 3), we will follow these steps:

### Step 1: Calculate the lengths of all sides
We will use the distance formula to find the lengths of the sides AB, BC, CD, and AD.

**Distance Formula:**
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

1. **Length of AB:**
\[ AB = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

2. **Length of BC:**
\[ BC = \sqrt{(7 - 3)^2 + (7 - 4)^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

3. **Length of CD:**
\[ CD = \sqrt{(7 - 4)^2 + (7 - 3)^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

4. **Length of AD:**
\[ AD = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 2: Check the slopes of opposite sides
Next, we will calculate the slopes of the sides to check if opposite sides are parallel.

**Slope Formula:**
The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

1. **Slope of AB:**
\[ m_{AB} = \frac{4 - 0}{3 - 0} = \frac{4}{3} \]

2. **Slope of CD:**
\[ m_{CD} = \frac{7 - 3}{7 - 4} = \frac{4}{3} \]

3. **Slope of AD:**
\[ m_{AD} = \frac{3 - 0}{4 - 0} = \frac{3}{4} \]

4. **Slope of BC:**
\[ m_{BC} = \frac{7 - 4}{7 - 3} = \frac{3}{4} \]

### Step 3: Analyze the results
- All sides are equal: \(AB = BC = CD = AD = 5\).
- Opposite sides are parallel: \(m_{AB} = m_{CD}\) and \(m_{AD} = m_{BC}\).

### Conclusion
Since all sides are equal and opposite sides are parallel, the quadrilateral ABCD is a **rhombus**.

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