https://file.aiquickdraw.com/mathbot/file/1726319110734j532c68i.jpg

Let's break down the problem in the image:

### Task 1 (a): 
We are given a triangle with vertices:
- \( A = (3, 0, -4) \)
- \( B = (-4, 0, 2) \)
- \( C = (3, 5, 5) \)

We need to calculate the lengths of the sides \( BC \), \( AC \), and \( AB \) using the distance formula in 3D:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

### Task 1 (b): 
We are given three points:
- \( A = (-3, 1, 4) \)
- \( B = (-1, -1, 2) \)
- \( C = (2, 0, 2) \)

We need to find the coordinates of point \( D = (D_x, D_y, D_z) \) such that \( ABCD \) forms a parallelogram. The condition for \( ABCD \) to be a parallelogram is that the vector \( \overrightarrow{AB} \) should be equal to the vector \( \overrightarrow{CD} \), meaning:
\[
D_x = C_x + (B_x - A_x)
\]
\[
D_y = C_y + (B_y - A_y)
\]
\[
D_z = C_z + (B_z - A_z)
\]

Let me compute the values for both parts.### Task 1 (a):
The lengths of the sides of the triangle are:
- \( |BC| = 9.110 \)
- \( |AC| = 10.296 \)
- \( |AB| = 9.220 \)

### Task 1 (b):
The coordinates of point \( D \) for the parallelogram \( ABCD \) are:
- \( D_x = 4 \)
- \( D_y = -2 \)
- \( D_z = 0 \)

Would you like further clarification on any step or more details?

Here are five additional related questions you could explore:
1. How is the distance formula derived for 3D space?
2. What conditions must be met for four points to form a parallelogram?
3. Can the coordinates of the centroid of the triangle \( ABC \) be found?
4. How do you calculate the area of a triangle in 3D using vectors?
5. How do different types of triangles affect the lengths of their sides?

**Tip:** When working with parallelograms, using vector notation helps simplify the process of finding unknown coordinates.

Given triangle with points A = (3, 0, -4), B = (-4, 0, 2), C = (3, 5, 5), find the side lengths of the triangle and determine the coordinates of D to form a parallelogram with points A = (-3, 1, 4), B = (-1, -1, 2), C = (2, 0, 2).

Solve a 3D geometry problem by calculating the side lengths of a triangle with vertices A = (3, 0, -4), B = (-4, 0, 2), and C = (3, 5, 5), and find the coordinates of point D for parallelogram ABCD with given points.

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