https://file.aiquickdraw.com/mathbot/file/1726196597158q0yrd4pb.jpg

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).

The points O(9, -3), P(6, 1), Q(2, -2), and R(5, -6) form quadrilateral OPQR. Plot the points then find the slopes and lengths, and identify the type of quadrilateral.

Determine the type of quadrilateral formed by points O(9, -3), P(6, 1), Q(2, -2), and R(5, -6) using coordinate geometry. This involves calculating slopes and lengths to confirm the quadrilateral is a rhombus. Suitable for students in Grades 8-10.

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