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The problem involves finding the slopes and lengths of the sides of quadrilateral DEFG and identifying the type of quadrilateral.

### Given points:
- \( D(0, 6) \)
- \( E(3, 4) \)
- \( F(8, 6) \)
- \( G(5, 9) \)

### Let's calculate the missing slope and length:

#### 1. **Slope of line \( DE \):**
The slope formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

For \( D(0, 6) \) and \( E(3, 4) \):
\[
\text{slope of } DE = \frac{4 - 6}{3 - 0} = \frac{-2}{3}
\]

So, the slope of \( DE \) is \( -\frac{2}{3} \).

#### 2. **Length of line \( DE \):**
The length formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
\text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

For \( D(0, 6) \) and \( E(3, 4) \):
\[
\text{length of } DE = \sqrt{(3 - 0)^2 + (4 - 6)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

So, the length of \( DE \) is \( \sqrt{13} \).

### Complete the missing information:
- Slope of \( DE = -\frac{2}{3} \)
- Length of \( DE = \sqrt{13} \)

### Identifying the quadrilateral:
Since the sides have different slopes and lengths, the quadrilateral DEFG is best described as a **generic quadrilateral**.

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Would you like a further explanation of any of these steps? Here are some related questions you could explore:

1. How do you calculate the slope between two points?
2. What is the relationship between slopes and parallelism?
3. Can the length formula be derived from the Pythagorean theorem?
4. What makes a quadrilateral classified as a parallelogram?
5. How do you identify congruent sides in a quadrilateral?

**Tip**: Understanding slope is key to recognizing parallel and perpendicular lines.

The points D(0, 6), E(3, 4), F(8, 6), and G(5, 9) form quadrilateral DEFG. Find the slopes and lengths of the sides and classify the quadrilateral.

Learn how to calculate the slopes and lengths of the sides of quadrilateral DEFG using the slope and distance formulas. This problem covers key concepts in coordinate geometry, including how to find slopes and lengths for a set of points on a graph. Suitable for students in Grades 9-10.

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