Whether the points (-1,-6,10) , (1,-3,4) , (-5,-1,1) and (-7,-4,7) form a rhombus

Answer

To determine whether the points (-1, -6, 10), (1, -3, 4), (-5, -1, 1), and (-7, -4, 7) form a rhombus, we need to check if all four sides of the quadrilateral formed by these points are equal. We will use the distance formula for three-dimensional space to find the lengths of the sides.

### Step 1: Identify the points
Let:
- A = (-1, -6, 10)
- B = (1, -3, 4)
- C = (-5, -1, 1)
- D = (-7, -4, 7)

### Step 2: Calculate the length of AB
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{(1 - (-1))^2 + (-3 - (-6))^2 + (4 - 10)^2} \]
\[ = \sqrt{(1 + 1)^2 + (-3 + 6)^2 + (4 - 10)^2} \]
\[ = \sqrt{(2)^2 + (3)^2 + (-6)^2} \]
\[ = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

### Step 3: Calculate the length of BC
\[ BC = \sqrt{((-5) - 1)^2 + ((-1) - (-3))^2 + (1 - 4)^2} \]
\[ = \sqrt{(-6)^2 + (2)^2 + (-3)^2} \]
\[ = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

### Step 4: Calculate the length of CD
\[ CD = \sqrt{((-7) - (-5))^2 + ((-4) - (-1))^2 + (7 - 1)^2} \]
\[ = \sqrt{(-2)^2 + (-3)^2 + (6)^2} \]
\[ = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

### Step 5: Calculate the length of DA
\[ DA = \sqrt{((-1) - (-7))^2 + ((-6) - (-4))^2 + (10 - 7)^2} \]
\[ = \sqrt{(6)^2 + (-2)^2 + (3)^2} \]
\[ = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

### Conclusion
Since all four sides are equal:
\[ AB = BC = CD = DA = 7 \]
The points (-1, -6, 10), (1, -3, 4), (-5, -1, 1), and (-7, -4, 7) form a rhombus.