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Let's tackle this problem step by step.

### (c) Given a quadrilateral with the vertices:
- \( A = (2, -5, 4) \)
- \( B = (-4, 2, -3) \)
- \( C = (-2, -2, -5) \)
- \( D = (4, -9, 2) \)

We are asked to determine whether the quadrilateral is:
- A parallelogram
- A rhombus
- A rectangle
- A square

#### Approach:
To determine the nature of the quadrilateral, we need to calculate the side lengths and diagonals using the distance formula between the points in 3D space:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

1. **Check for parallelogram**: A quadrilateral is a parallelogram if opposite sides are equal.
2. **Check for rhombus**: All sides must be equal in length.
3. **Check for rectangle**: Opposite sides are equal and diagonals are also equal.
4. **Check for square**: All sides and diagonals are equal.

Let's start by calculating the lengths of all sides and diagonals.

---

### (d) Determine all values of parameter \( t \) so that the points form an isosceles triangle with sides \( AC \) and \( AB \).

Points:
- \( A = (3, 3, 10) \)
- \( B = (-4, -3, 3) \)
- \( C = (3, 2, t) \)

For an isosceles triangle, the lengths of two sides must be equal:
\[
AC = AB
\]

We will set the distances \( AC \) and \( AB \) equal and solve for \( t \).

---

I'll calculate these values for you now.### (c) Quadrilateral Analysis

We have the following distances (side lengths and diagonals):
- \( AB = 11.576 \)
- \( BC = 4.899 \)
- \( CD = 11.576 \)
- \( DA = 4.899 \)
- Diagonal \( AC = 10.296 \)
- Diagonal \( BD = 14.491 \)

Now, let's check the conditions for each type of quadrilateral:

1. **Parallelogram**: Opposite sides are equal.  
   - \( AB = CD \) and \( BC = DA \), so **yes**, it is a parallelogram.
   
2. **Rhombus**: All sides must be equal.  
   - The sides are not all equal (\( AB \neq BC \)), so **no**, it is not a rhombus.
   
3. **Rectangle**: Opposite sides are equal, and diagonals must be equal.  
   - The diagonals are not equal (\( AC \neq BD \)), so **no**, it is not a rectangle.
   
4. **Square**: All sides and diagonals must be equal.  
   - Neither the sides nor the diagonals are all equal, so **no**, it is not a square.

### (d) Solving for \( t \)

The possible values of \( t \) that make the triangle \( ABC \) isosceles (i.e., \( AC = AB \)) are:
\[
t_1 = -11.576, \quad t_2 = 11.576
\]

### Summary of Answers:
- Parallelogram: **Yes**
- Rhombus: **No**
- Rectangle: **No**
- Square: **No**
- \( t_1 = -11.576 \), \( t_2 = 11.576 \)

Would you like any further details or clarifications? Here are 5 related questions to deepen understanding:

1. How is the distance formula in 3D space derived?
2. How can you determine whether diagonals are perpendicular in 3D space?
3. What are the necessary and sufficient conditions for a quadrilateral to be a rhombus?
4. Can an isosceles triangle exist with any set of 3 points in 3D space?
5. How does the use of parameters like \( t \) change the flexibility of triangle formation in geometric problems?

**Tip**: Understanding the properties of geometric shapes in 3D, such as the relationships between sides and diagonals, can make complex problems much simpler.

Given a quadrilateral with vertices A = (2, -5, 4), B = (-4, 2, -3), C = (-2, -2, -5), D = (4, -9, 2), determine whether it is a parallelogram, rhombus, rectangle, or square. Also, determine the values of parameter t such that points A = (3, 3, 10), B = (-4, -3, 3), and C = (3, 2, t) form an isosceles triangle with sides AC and AB.

Learn how to analyze a 3D quadrilateral with vertices A = (2, -5, 4), B = (-4, 2, -3), C = (-2, -2, -5), D = (4, -9, 2) to determine if it is a parallelogram, rhombus, rectangle, or square. Additionally, solve for parameter t in an isosceles triangle with sides AC and AB, where A = (3, 3, 10), B = (-4, -3, 3), and C = (3, 2, t). Suitable for students in Grades 10-12.

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