The points `A(1,- 6,10), B(-1, -3,4), C(5, - 1,1)` and `D(7,-4,7)` are the vertices of a

Options

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

Answer

To determine the shape formed by the points A(1, -6, 10), B(-1, -3, 4), C(5, -1, 1), and D(7, -4, 7), we will calculate the distances between each pair of points and analyze the results.

### Step-by-Step Solution:

1. **Calculate the distance AB:**
- Use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
- For points A(1, -6, 10) and B(-1, -3, 4):
\[ AB = \sqrt{((-1) - 1)^2 + ((-3) - (-6))^2 + (4 - 10)^2} \]
\[ = \sqrt{(-2)^2 + (3)^2 + (-6)^2} \]
\[ = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

2. **Calculate the distance BC:**
- For points B(-1, -3, 4) and C(5, -1, 1):
\[ BC = \sqrt{(5 - (-1))^2 + ((-1) - (-3))^2 + (1 - 4)^2} \]
\[ = \sqrt{(6)^2 + (2)^2 + (-3)^2} \]
\[ = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

3. **Calculate the distance CD:**
- For points C(5, -1, 1) and D(7, -4, 7):
\[ CD = \sqrt{(7 - 5)^2 + ((-4) - (-1))^2 + (7 - 1)^2} \]
\[ = \sqrt{(2)^2 + (-3)^2 + (6)^2} \]
\[ = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

4. **Calculate the distance DA:**
- For points D(7, -4, 7) and A(1, -6, 10):
\[ DA = \sqrt{(1 - 7)^2 + ((-6) - (-4))^2 + (10 - 7)^2} \]
\[ = \sqrt{(-6)^2 + (-2)^2 + (3)^2} \]
\[ = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

5. **Summary of distances:**
- \( AB = 7 \)
- \( BC = 7 \)
- \( CD = 7 \)
- \( DA = 7 \)

6. **Calculate the diagonals AC and BD:**
- For diagonal AC:
\[ AC = \sqrt{(5 - 1)^2 + ((-1) - (-6))^2 + (1 - 10)^2} \]
\[ = \sqrt{(4)^2 + (5)^2 + (-9)^2} \]
\[ = \sqrt{16 + 25 + 81} = \sqrt{122} \]

- For diagonal BD:
\[ BD = \sqrt{(7 - (-1))^2 + ((-4) - (-3))^2 + (7 - 4)^2} \]
\[ = \sqrt{(8)^2 + (-1)^2 + (3)^2} \]
\[ = \sqrt{64 + 1 + 9} = \sqrt{74} \]

7. **Conclusion:**
- All sides are equal, \( AB = BC = CD = DA = 7 \).
- The diagonals \( AC \) and \( BD \) are not equal.
- Therefore, the quadrilateral formed by points A, B, C, and D is a **Rombus**.