A pair of points is given. $$ (0,-2),(7,0) $$

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Certainly! Given the two points:

$$
(0, -2) \quad \text{and} \quad (7, 0)
$$

there are several properties and calculations we can explore, such as finding the **slope** of the line connecting them, the **equation of the line**, the **distance** between the points, and the **midpoint** of the segment joining them. Here's a comprehensive breakdown:

---

### 1. **Slope of the Line ($$m$$)**

The slope measures the steepness of the line connecting two points and is calculated using the formula:

$$
m = \frac{{y_2 - y_1}}{{x_2 - x_1}}
$$

**Given:**
$$
(x_1, y_1) = (0, -2), \quad (x_2, y_2) = (7, 0)
$$

**Calculation:**
$$
m = \frac{{0 - (-2)}}{{7 - 0}} = \frac{2}{7}
$$

**So, the slope $$m = \frac{2}{7}$$.**

---

### 2. **Equation of the Line**

With the slope known, we can express the equation of the line in **slope-intercept form** ($$y = mx + b$$) or **point-slope form**.

#### a. **Slope-Intercept Form ($$y = mx + b$$)**

**We have:**
$$
m = \frac{2}{7}
$$

**To find $$b$$ (y-intercept), use one of the points. Let's use $$(0, -2)$$:**
$$
-2 = \frac{2}{7}(0) + b \implies b = -2
$$

**Equation:**
$$
y = \frac{2}{7}x - 2
$$

#### b. **Point-Slope Form**
$$
y - y_1 = m(x - x_1)
$$

Using point $$(0, -2)$$:

$$
y - (-2) = \frac{2}{7}(x - 0) \implies y + 2 = \frac{2}{7}x
$$

Which simplifies to the same slope-intercept form:
$$
y = \frac{2}{7}x - 2
$$

---

### 3. **Distance Between the Points**

The distance ($$d$$) between two points in a plane is given by the formula:

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

**Calculation:**
$$
d = \sqrt{(7 - 0)^2 + (0 - (-2))^2} = \sqrt{7^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53}
$$

**So, the distance $$d = \sqrt{53}$$ units.** Approximately, $$ \sqrt{53} \approx 7.28 $$ units.

---

### 4. **Midpoint of the Segment**

The midpoint ($$M$$) between two points is the average of their respective coordinates.

$$
M = \left( \frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2} \right)
$$

**Calculation:**
$$
M = \left( \frac{{0 + 7}}{2}, \frac{{-2 + 0}}{2} \right) = \left( \frac{7}{2}, -1 \right) = (3.5, -1)
$$

**So, the midpoint is at $$(3.5, -1)$$.**

---

### 5. **Additional Insights**

- **Vertical or Horizontal Line Check:**
  - **Vertical Line:** If $$x_1 = x_2$$, the line is vertical.
  - **Horizontal Line:** If $$y_1 = y_2$$, the line is horizontal.
  
  In this case, since both $$x$$-coordinates ($$0$$ and $$7$$) and $$y$$-coordinates ($$-2$$ and $$0$$) are different, the line is neither vertical nor horizontal.

- **Slope Nature:**
  - A positive slope ($$\frac{2}{7}$$ in this case) indicates that as $$x$$ increases, $$y$$ also increases.

---

### 6. **Graphical Representation**

Plotting the points and the line can provide a visual understanding:

1. **Plot the Points:**
   - $$(0, -2)$$: Located on the y-axis, 2 units below the origin.
   - $$(7, 0)$$: Located on the x-axis, 7 units to the right of the origin.

2. **Draw the Line:**
   - Connect the two points with a straight line, which will have a gentle upward slope since $$m = \frac{2}{7}$$.

3. **Mark the Midpoint:**
   - At $$(3.5, -1)$$, the midpoint lies halfway between the two points.

---

If you have a specific question or need further assistance related to these points, feel free to ask!
The two points are (0, -2) and (7, 0). The slope of the line connecting them is 2/7. The equation of the line is y = (2/7)x - 2. The distance between the points is √53 units, which is approximately 7.28 units. The midpoint of the segment joining them is (3.5, -1).

A pair of points is given. \[ (0,-2),(7,0) \]

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