The points (5, -4, 2), (4, -3, 1), (7, -6, 4) and
(8, -7, 5) are the vertices of a

Options

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

Answer

To determine what the points (5, -4, 2), (4, -3, 1), (7, -6, 4), and (8, -7, 5) represent in three-dimensional space, we can follow these steps:

### Step 1: Assign the Points
Let:
- Point A = (5, -4, 2)
- Point B = (4, -3, 1)
- Point C = (7, -6, 4)
- Point D = (8, -7, 5)

### Step 2: Calculate the Length of AB
Using the distance formula in 3D:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
\[ AB = \sqrt{(4 - 5)^2 + (-3 + 4)^2 + (1 - 2)^2} \]
\[ = \sqrt{(-1)^2 + (1)^2 + (-1)^2} \]
\[ = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 3: Calculate the Length of BC
\[ BC = \sqrt{(7 - 4)^2 + (-6 + 3)^2 + (4 - 1)^2} \]
\[ = \sqrt{(3)^2 + (-3)^2 + (3)^2} \]
\[ = \sqrt{9 + 9 + 9} = \sqrt{27} \]

### Step 4: Calculate the Length of CD
\[ CD = \sqrt{(8 - 7)^2 + (-7 + 6)^2 + (5 - 4)^2} \]
\[ = \sqrt{(1)^2 + (-1)^2 + (1)^2} \]
\[ = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 5: Calculate the Length of DA
\[ DA = \sqrt{(5 - 8)^2 + (-4 + 7)^2 + (2 - 5)^2} \]
\[ = \sqrt{(-3)^2 + (3)^2 + (-3)^2} \]
\[ = \sqrt{9 + 9 + 9} = \sqrt{27} \]

### Step 6: Calculate the Length of the Diagonal AC
\[ AC = \sqrt{(7 - 5)^2 + (-6 + 4)^2 + (4 - 2)^2} \]
\[ = \sqrt{(2)^2 + (-2)^2 + (2)^2} \]
\[ = \sqrt{4 + 4 + 4} = \sqrt{12} \]

### Step 7: Calculate the Length of the Diagonal BD
\[ BD = \sqrt{(8 - 4)^2 + (-7 + 3)^2 + (5 - 1)^2} \]
\[ = \sqrt{(4)^2 + (-4)^2 + (4)^2} \]
\[ = \sqrt{16 + 16 + 16} = \sqrt{48} \]

### Step 8: Analyze the Lengths
From our calculations:
- Length of AB = Length of CD = \( \sqrt{3} \)
- Length of AD = Length of BC = \( \sqrt{27} \)
- Length of AC = \( \sqrt{12} \)
- Length of BD = \( \sqrt{48} \)

Since opposite sides are equal (AB = CD and AD = BC) but the diagonals are not equal (AC ≠ BD), the figure formed by these points is a parallelogram.

### Conclusion
The points (5, -4, 2), (4, -3, 1), (7, -6, 4), and (8, -7, 5) represent the vertices of a parallelogram in three-dimensional space.

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