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.