To show that the points $(0, 5)$, $(3, -4)$, $(8, 0)$, and $(3, -1)$ are the vertices of a parallelogram, we need to verify that either:

1. Opposite sides are parallel, or
2. Diagonals bisect each other.

Step 1: Label the points

Let the points be labeled as:

• $A(0, 5)$
• $B(3, -4)$
• $C(8, 0)$
• $D(3, -1)$

Step 2: Check if opposite sides are parallel

To determine if opposite sides are parallel, we need to calculate the slopes of opposite sides.

Slope of line $AB$:

The slope of the line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$\text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 5}{3 - 0} = \frac{-9}{3} = -3$

Slope of line $CD$:

$\text{slope of } CD = \frac{-1 - 0}{3 - 8} = \frac{-1}{-5} = \frac{1}{5}$

So, lines $AB$ and $CD$ are not parallel.

Slope of line $BC$:

$\text{slope of } BC = \frac{0 - (-4)}{8 - 3} = \frac{4}{5}$

Slope of line $AD$:

$\text{slope of } AD = \frac{-1 - 5}{3 - 0} = \frac{-6}{3} = -2$

So, lines $BC$ and $AD$ are also not parallel.

Step 3: Check if diagonals bisect each other

The diagonals of a parallelogram bisect each other, meaning their midpoints should be the same.

Midpoint of diagonal $AC$:

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$\text{Midpoint of } AC = \left(\frac{0 + 8}{2}, \frac{5 + 0}{2}\right) = (4, 2.5)$

Midpoint of diagonal $BD$:

$\text{Midpoint of } BD = \left(\frac{3 + 3}{2}, \frac{-4 + (-1)}{2}\right) = (3, -2.5)$

Since the midpoints of diagonals $AC$ and $BD$ are not the same, the diagonals do not bisect each other.

Conclusion:

The points do not form a parallelogram because neither the opposite sides are parallel nor do the diagonals bisect each other.

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Would you like further details or clarifications?

Here are five related questions for further exploration:

1. What conditions must be satisfied for points to form a rectangle?
2. How can we verify if four points form a rhombus?
3. What is the formula for calculating the area of a parallelogram given its vertices?
4. How do the slopes of diagonals in a parallelogram relate to its shape?
5. Can we classify quadrilaterals based on side lengths and angles?

Tip: When working with coordinates, always double-check slope calculations to avoid errors.