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.