To determine the form of the figure represented by the points \( A(5, -5, 2) \), \( B(4, -3, 1) \), \( C(7, -6, 4) \), and \( D(8, -7, 5) \), we will calculate the lengths of the sides and the diagonals of the quadrilateral formed by these points.
### Step 1: Calculate the length of side AB
The length \( AB \) can be calculated 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 points \( A \) and \( B \):
\[
AB = \sqrt{(4 - 5)^2 + (-3 + 5)^2 + (1 - 2)^2} = \sqrt{(-1)^2 + (2)^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}
\]
### Step 2: Calculate the length of side BC
Using the same distance formula for points \( B \) and \( C \):
\[
BC = \sqrt{(7 - 4)^2 + (-6 + 3)^2 + (4 - 1)^2} = \sqrt{(3)^2 + (-3)^2 + (3)^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3}
\]
### Step 3: Calculate the length of side CD
Now, calculate the length \( CD \):
\[
CD = \sqrt{(8 - 7)^2 + (-7 + 6)^2 + (5 - 4)^2} = \sqrt{(1)^2 + (-1)^2 + (1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}
\]
### Step 4: Calculate the length of side DA
Next, calculate the length \( DA \):
\[
DA = \sqrt{(5 - 8)^2 + (-5 + 7)^2 + (2 - 5)^2} = \sqrt{(-3)^2 + (2)^2 + (-3)^2} = \sqrt{9 + 4 + 9} = \sqrt{22}
\]
### Step 5: Calculate the length of diagonal AC
Now, calculate the diagonal \( AC \):
\[
AC = \sqrt{(7 - 5)^2 + (-6 + 5)^2 + (4 - 2)^2} = \sqrt{(2)^2 + (-1)^2 + (2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3
\]
### Step 6: Calculate the length of diagonal BD
Finally, calculate the diagonal \( BD \):
\[
BD = \sqrt{(8 - 4)^2 + (-7 + 3)^2 + (5 - 1)^2} = \sqrt{(4)^2 + (-4)^2 + (4)^2} = \sqrt{16 + 16 + 16} = \sqrt{48} = 4\sqrt{3}
\]
### Step 7: Analyze the lengths
We have:
- \( AB = \sqrt{6} \)
- \( BC = 3\sqrt{3} \)
- \( CD = \sqrt{3} \)
- \( DA = \sqrt{22} \)
- \( AC = 3 \)
- \( BD = 4\sqrt{3} \)
To determine if the figure is a parallelogram, we check if opposite sides are equal:
- \( AB \) and \( CD \) are not equal.
- \( BC \) and \( DA \) are not equal.
- However, we can observe that \( AC \) and \( BD \) are not equal either.
### Conclusion
Since opposite sides are not equal, the figure formed by the points \( A \), \( B \), \( C \), and \( D \) is not a parallelogram.
