Which of the following figures is generated using the four points `(1,3),(1,9),(1+ 3sqrt(3),12)` and `(1+3sqrt(3),6)` taken in the order as mentioned ?

Options

A. Rectangle
B. Rhombus
C. Square
D. Trapezium

Answer

To determine the figure generated by the points \( (1,3), (1,9), (1 + 3\sqrt{3}, 12), (1 + 3\sqrt{3}, 6) \), we will analyze the coordinates step by step.

### Step 1: Plot the Points
We start by plotting the given points on a Cartesian coordinate system.

1. **Point A**: \( (1, 3) \)
2. **Point B**: \( (1, 9) \)
3. **Point C**: \( (1 + 3\sqrt{3}, 12) \)
4. **Point D**: \( (1 + 3\sqrt{3}, 6) \)

### Step 2: Identify Coordinates
- The x-coordinates of points A and B are both 1, indicating that they lie on the vertical line \( x = 1 \).
- The x-coordinates of points C and D are \( 1 + 3\sqrt{3} \).

### Step 3: Calculate Lengths of Sides
Next, we calculate the lengths of the sides formed by these points.

1. **Length AB**:
\[ AB = |y_B - y_A| = |9 - 3| = 6 \]

2. **Length CD**:
\[ CD = |y_C - y_D| = |12 - 6| = 6 \]

3. **Length AD**:
- The distance between points A and D can be calculated using the x-coordinates and y-coordinates:
\[ AD = \sqrt{(x_D - x_A)^2 + (y_D - y_A)^2} = \sqrt{((1 + 3\sqrt{3}) - 1)^2 + (6 - 3)^2} \]
\[ = \sqrt{(3\sqrt{3})^2 + 3^2} = \sqrt{27 + 9} = \sqrt{36} = 6 \]

4. **Length BC**:
- Similarly, we calculate the distance between points B and C:
\[ BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{((1 + 3\sqrt{3}) - 1)^2 + (12 - 9)^2} \]
\[ = \sqrt{(3\sqrt{3})^2 + 3^2} = \sqrt{27 + 9} = \sqrt{36} = 6 \]

### Step 4: Check Parallelism
- **Lines AB and CD**: Both are vertical lines (same x-coordinates).
- **Lines AD and BC**: Both have the same length and are not vertical, indicating they are parallel.

### Step 5: Conclusion
Since all sides are equal (AB = BC = CD = AD = 6) and opposite sides are parallel, the figure formed by these points is a **rhombus**.

### Final Answer
The figure generated by the points is a **rhombus**.

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