Let's go step by step to solve the problem based on the image.

Points:

• $O(9, -3)$
• $P(6, 1)$
• $Q(2, -2)$
• $R(5, -6)$

Part 1: Slopes

We will calculate the slope of each pair of points using the slope formula: $\text{slope of line} = \frac{y_2 - y_1}{x_2 - x_1}$

1. Slope of $OP$:

Points $O(9, -3)$ and $P(6, 1)$ $\text{slope of } OP = \frac{1 - (-3)}{6 - 9} = \frac{4}{-3} = -\frac{4}{3}$

2. Slope of $PQ$:

Points $P(6, 1)$ and $Q(2, -2)$ $\text{slope of } PQ = \frac{-2 - 1}{2 - 6} = \frac{-3}{-4} = \frac{3}{4}$

3. Slope of $QR$:

Points $Q(2, -2)$ and $R(5, -6)$ $\text{slope of } QR = \frac{-6 - (-2)}{5 - 2} = \frac{-4}{3} = -\frac{4}{3}$

4. Slope of $RO$:

Points $R(5, -6)$ and $O(9, -3)$ $\text{slope of } RO = \frac{-3 - (-6)}{9 - 5} = \frac{3}{4}$

Part 2: Lengths

We will calculate the lengths using the distance formula: $\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Length of $OP$:

$\text{length of } OP = \sqrt{(6 - 9)^2 + (1 - (-3))^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

2. Length of $PQ$:

$\text{length of } PQ = \sqrt{(2 - 6)^2 + (-2 - 1)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

3. Length of $QR$:

$\text{length of } QR = \sqrt{(5 - 2)^2 + (-6 - (-2))^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

4. Length of $RO$:

$\text{length of } RO = \sqrt{(9 - 5)^2 + (-3 - (-6))^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Part 3: Identifying the Quadrilateral

Since all sides are equal in length and opposite sides are parallel (as indicated by equal slopes), the quadrilateral is a rhombus.

Final Answer:

• Slope of $OP = -\frac{4}{3}$
• Slope of $PQ = \frac{3}{4}$
• Slope of $QR = -\frac{4}{3}$
• Slope of $RO = \frac{3}{4}$
• Length of $OP = 5$
• Length of $PQ = 5$
• Length of $QR = 5$
• Length of $RO = 5$
• Quadrilateral OPQR is best described as a rhombus.

Would you like more details or have any questions?

Related Questions:

1. How do you prove a quadrilateral is a square using slopes and lengths?
2. What is the relationship between parallel lines and their slopes?
3. How can you calculate the area of a rhombus given its diagonals?
4. How does the distance formula relate to the Pythagorean theorem?
5. What distinguishes a rhombus from a parallelogram?

Tip:

For any quadrilateral, verifying both the side lengths and the angles formed by slopes helps confirm the specific type of quadrilateral (e.g., square, rhombus, or rectangle).