https://file.aiquickdraw.com/mathbot/file/1730753397765jucvgl4p.jpg

The image you provided displays two points:  
- Point 1: \((-19, 16)\)  
- Point 2: \((-15, -17)\)  

These coordinates are likely part of a problem that involves calculating properties such as the **distance** between the points, the **slope** of the line passing through them, or the **midpoint**.

Let's solve for each of these:

### 1. **Distance between the points**
The formula for the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Plugging in \((x_1, y_1) = (-19, 16)\) and \((x_2, y_2) = (-15, -17)\):
\[
d = \sqrt{(-15 + 19)^2 + (-17 - 16)^2} = \sqrt{(4)^2 + (-33)^2}
\]
\[
d = \sqrt{16 + 1089} = \sqrt{1105} \approx 33.24
\]

### 2. **Slope of the line passing through the points**
The slope \(m\) between two points is calculated by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using \((x_1, y_1) = (-19, 16)\) and \((x_2, y_2) = (-15, -17)\):
\[
m = \frac{-17 - 16}{-15 + 19} = \frac{-33}{4} = -8.25
\]

### 3. **Midpoint of the line segment joining the points**
The midpoint \(M\) of two points is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Calculating with \((x_1, y_1) = (-19, 16)\) and \((x_2, y_2) = (-15, -17)\):
\[
M = \left( \frac{-19 + (-15)}{2}, \frac{16 + (-17)}{2} \right) = \left( \frac{-34}{2}, \frac{-1}{2} \right)
\]
\[
M = (-17, -0.5)
\]

### Summary of Results
- **Distance** ≈ 33.24
- **Slope** = -8.25
- **Midpoint** = \((-17, -0.5)\)

Would you like more details on any part of the solution, or do you have any specific questions?

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Here are some additional related questions:
1. How do we find the equation of the line passing through these points?
2. How does the negative slope affect the direction of the line?
3. Can you derive the midpoint formula from basic principles?
4. What would change if one of the points was in a different quadrant?
5. How would we calculate the angle of inclination of the line formed by these points?

**Tip:** Always recheck your signs when plugging coordinates into formulas, especially with negative values.

Given two points (-19, 16) and (-15, -17), calculate the distance, slope, and midpoint.

Learn how to calculate the distance, slope, and midpoint between two points (-19, 16) and (-15, -17). This problem demonstrates key concepts in coordinate geometry using distance, slope, and midpoint formulas, suitable for Grades 9-10.

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