show that the points (0,5) (3,-4) (8,0) and (3,-1) are the vertices of a parallelogram

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.

---

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.

Show that the points (0,5) (3,-4) (8,0) and (3,-1) are the vertices of a parallelogram.

Learn how to verify if the points (0,5), (3,-4), (8,0), and (3,-1) form a parallelogram using slope and midpoint calculations. This problem applies coordinate geometry concepts and explores properties of parallelograms. Suitable for students in Grades 9-10.

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