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To determine whether the points \( A(4,1) \), \( B(-1,-4) \), \( C(3,3) \), and \( D(8,8) \) form a parallelogram, we can use the following approach:

### Step 1: Calculate the slopes of opposite sides
For the quadrilateral to be a parallelogram, opposite sides must have the same slope. We will calculate the slopes of \( AB \), \( CD \), \( BC \), and \( AD \).

The formula for slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

- **Slope of \( AB \):**
\[
\text{slope of } AB = \frac{-4 - 1}{-1 - 4} = \frac{-5}{-5} = 1
\]

- **Slope of \( CD \):**
\[
\text{slope of } CD = \frac{8 - 3}{8 - 3} = \frac{5}{5} = 1
\]
  
- **Slope of \( BC \):**
\[
\text{slope of } BC = \frac{3 - (-4)}{3 - (-1)} = \frac{3 + 4}{3 + 1} = \frac{7}{4}
\]

- **Slope of \( AD \):**
\[
\text{slope of } AD = \frac{8 - 1}{8 - 4} = \frac{7}{4}
\]

### Step 2: Conclusion
Since the slopes of opposite sides \( AB \) and \( CD \) are equal, and the slopes of opposite sides \( BC \) and \( AD \) are equal, the points form a parallelogram.

Thus, the four points **do** form a parallelogram.

Would you like more details or have any further questions?

Here are some related questions:
1. How do you verify if a quadrilateral is a parallelogram using distance?
2. Can this method be applied to determine if a quadrilateral is a rectangle?
3. How can you calculate the area of a parallelogram given four vertices?
4. What is the difference between slopes in a parallelogram and a trapezoid?
5. How do you confirm if a set of points form a square?

**Tip:** In geometry, opposite sides being parallel (equal slopes) is a key criterion for identifying a parallelogram.

Determine whether or not the points A(4,1), B(-1,-4), C(3,3), and D(8,8) form a parallelogram.

Learn how to check if the points A(4,1), B(-1,-4), C(3,3), and D(8,8) form a parallelogram by using slopes of opposite sides. This problem involves applying the slope formula to ensure that opposite sides are parallel.

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