To analyze the line segment CD with endpoints C(-8, -6) and D(-4, 10), we can find several key properties:

1. Length of CD

The length of the line segment can be calculated using the distance formula: $\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates of C(-8, -6) and D(-4, 10): $\text{Length} = \sqrt{(-4 - (-8))^2 + (10 - (-6))^2} = \sqrt{(4)^2 + (16)^2} = \sqrt{16 + 256} = \sqrt{272} = 4\sqrt{17} \approx 16.49$

2. Midpoint of CD

The midpoint of the line segment can be found using the midpoint formula: $\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ Substituting the coordinates of C(-8, -6) and D(-4, 10): $\text{Midpoint} = \left(\frac{-8 + (-4)}{2}, \frac{-6 + 10}{2}\right) = \left(\frac{-12}{2}, \frac{4}{2}\right) = (-6, 2)$

3. Slope of CD

The slope of the line segment is calculated using the slope formula: $\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the coordinates of C(-8, -6) and D(-4, 10): $\text{Slope} = \frac{10 - (-6)}{-4 - (-8)} = \frac{16}{4} = 4$

Summary

• Length of CD: $4\sqrt{17}$ or approximately 16.49 units
• Midpoint of CD: (-6, 2)
• Slope of CD: 4

Would you like any further details or have any questions?

Here are some related questions:

1. How do you find the equation of the line that passes through points C and D?
2. What is the perpendicular slope to the line CD?
3. How do you calculate the area of a triangle formed with another point and this line segment?
4. What are the x and y intercepts of the line through CD?
5. How does the distance from a point to a line compare with the segment length?

Tip: The distance formula is essentially the Pythagorean Theorem applied to the difference in coordinates.