Given the points A(-2, 3), B(6, 1), C(5,-2) and D(-3, 0), show that ABCD is a parallelogram.

To show that the quadrilateral ABCD is a parallelogram, we need to verify that either:

1. Both pairs of opposite sides are parallel, or
2. Both pairs of opposite sides are equal in length.

I'll use the second method here: showing that opposite sides are equal in length.

### Step 1: Distance formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 2: Find the lengths of sides AB, BC, CD, and DA.

#### Length of AB:
The coordinates of A and B are \((-2, 3)\) and \((6, 1)\), respectively. Applying the distance formula:

\[
AB = \sqrt{(6 - (-2))^2 + (1 - 3)^2} = \sqrt{(6 + 2)^2 + (-2)^2} = \sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68}
\]

\[
AB = \sqrt{68} \approx 8.25
\]

#### Length of CD:
The coordinates of C and D are \((5, -2)\) and \((-3, 0)\), respectively. Applying the distance formula:

\[
CD = \sqrt{(5 - (-3))^2 + (-2 - 0)^2} = \sqrt{(5 + 3)^2 + (-2)^2} = \sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68}
\]

\[
CD = \sqrt{68} \approx 8.25
\]

So, \(AB = CD\).

#### Length of AD:
The coordinates of A and D are \((-2, 3)\) and \((-3, 0)\), respectively. Applying the distance formula:

\[
AD = \sqrt{(-3 - (-2))^2 + (0 - 3)^2} = \sqrt{(-3 + 2)^2 + (-3)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

\[
AD = \sqrt{10} \approx 3.16
\]

#### Length of BC:
The coordinates of B and C are \((6, 1)\) and \((5, -2)\), respectively. Applying the distance formula:

\[
BC = \sqrt{(5 - 6)^2 + (-2 - 1)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

\[
BC = \sqrt{10} \approx 3.16
\]

So, \(AD = BC\).

### Step 3: Conclusion

Since opposite sides \(AB = CD\) and \(AD = BC\), the quadrilateral ABCD is a **parallelogram**.

---

Would you like any further details or have questions? Here are some additional topics to explore:

1. How do you prove a parallelogram is a rhombus?
2. How can you calculate the area of parallelogram

Learn how to prove that the quadrilateral ABCD, with given coordinates, is a parallelogram using coordinate geometry concepts. The solution involves calculating the lengths of opposite sides using the distance formula and verifying they are equal. Suitable for students in Grades 9-10.

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