The points (5, -1,1)(7,4,7), (1, -6, 10) and (-1, -3, 4) are the vertices of a

Options

A. trapezium
B. rhombus
C. rectangle
D. None of these

Answer

To determine the type of quadrilateral formed by the points \( A(5, -1, 1) \), \( B(7, 4, 7) \), \( C(1, -6, 10) \), and \( D(-1, -3, 4) \), we will analyze the distances between the points and their direction ratios.

### Step 1: Calculate Direction Ratios
First, we calculate the direction ratios for the line segments \( AB \), \( BC \), \( CD \), and \( DA \).

1. **Direction Ratio of \( AB \)**:
\[ AB: (7 - 5, 4 - (-1), 7 - 1) = (2, 5, 6) \]

2. **Direction Ratio of \( BC \)**:
\[ BC: (1 - 7, -6 - 4, 10 - 7) = (-6, -10, 3) \]

3. **Direction Ratio of \( CD \)**:
\[ CD: (-1 - 1, -3 - (-6), 4 - 10) = (-2, 3, -6) \]

4. **Direction Ratio of \( DA \)**:
\[ DA: (5 - (-1), -1 - (-3), 1 - 4) = (6, 2, -3) \]

### Step 2: Check for Parallelism
Next, we check if any pairs of opposite sides are parallel by comparing their direction ratios.

- **Comparing \( AB \) and \( CD \)**:
\[ (2, 5, 6) \quad \text{and} \quad (-2, 3, -6) \]
These are not scalar multiples, so \( AB \) is not parallel to \( CD \).

- **Comparing \( BC \) and \( DA \)**:
\[ (-6, -10, 3) \quad \text{and} \quad (6, 2, -3) \]
These are also not scalar multiples, so \( BC \) is not parallel to \( DA \).

### Step 3: Calculate Distances
Now we calculate the lengths of all sides using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

1. **Distance \( AB \)**:
\[ AB = \sqrt{(7 - 5)^2 + (4 - (-1))^2 + (7 - 1)^2} = \sqrt{2^2 + 5^2 + 6^2} = \sqrt{4 + 25 + 36} = \sqrt{65} \]

2. **Distance \( BC \)**:
\[ BC = \sqrt{(1 - 7)^2 + (-6 - 4)^2 + (10 - 7)^2} = \sqrt{(-6)^2 + (-10)^2 + 3^2} = \sqrt{36 + 100 + 9} = \sqrt{145} \]

3. **Distance \( CD \)**:
\[ CD = \sqrt{(-1 - 1)^2 + (-3 - (-6))^2 + (4 - 10)^2} = \sqrt{(-2)^2 + 3^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

4. **Distance \( DA \)**:
\[ DA = \sqrt{(5 - (-1))^2 + (-1 - (-3))^2 + (1 - 4)^2} = \sqrt{(6)^2 + (2)^2 + (-3)^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

### Step 4: Analyze the Results
From the calculations:
- \( AB = \sqrt{65} \)
- \( BC = \sqrt{145} \)
- \( CD = 7 \)
- \( DA = 7 \)

Since \( AB \) and \( BC \) are not equal to \( CD \) and \( DA \), the quadrilateral does not have equal opposite sides, ruling out a rectangle or rhombus.

### Conclusion
Since neither pairs of opposite sides are parallel nor are all sides equal, the quadrilateral formed by the points \( A, B, C, D \) is not a trapezium, rhombus, rectangle, or any standard quadrilateral. Therefore, the answer is **none of these options**.