16. $$ N(-4,5), O(6,0), P(3,-6), Q(-7,-1) $$; Distance Formula

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Sure! Let's calculate the distances between each pair of points $$ N(-4,5) $$, $$ O(6,0) $$, $$ P(3,-6) $$, and $$ Q(-7,-1) $$ using the **Distance Formula**.

### **Distance Formula**
The distance $$ d $$ between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by:
$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

### **Calculations**

We'll calculate the distance for each pair:

1. **Distance between $$ N(-4,5) $$ and $$ O(6,0) $$:**
   $$
   \begin{align*}
   \Delta x &= 6 - (-4) = 10 \
   \Delta y &= 0 - 5 = -5 \
   d &= \sqrt{10^2 + (-5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \approx 11.18
   \end{align*}
   $$

2. **Distance between $$ N(-4,5) $$ and $$ P(3,-6) $$:**
   $$
   \begin{align*}
   \Delta x &= 3 - (-4) = 7 \
   \Delta y &= -6 - 5 = -11 \
   d &= \sqrt{7^2 + (-11)^2} = \sqrt{49 + 121} = \sqrt{170} \approx 13.04
   \end{align*}
   $$

3. **Distance between $$ N(-4,5) $$ and $$ Q(-7,-1) $$:**
   $$
   \begin{align*}
   \Delta x &= -7 - (-4) = -3 \
   \Delta y &= -1 - 5 = -6 \
   d &= \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \approx 6.71
   \end{align*}
   $$

4. **Distance between $$ O(6,0) $$ and $$ P(3,-6) $$:**
   $$
   \begin{align*}
   \Delta x &= 3 - 6 = -3 \
   \Delta y &= -6 - 0 = -6 \
   d &= \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \approx 6.71
   \end{align*}
   $$

5. **Distance between $$ O(6,0) $$ and $$ Q(-7,-1) $$:**
   $$
   \begin{align*}
   \Delta x &= -7 - 6 = -13 \
   \Delta y &= -1 - 0 = -1 \
   d &= \sqrt{(-13)^2 + (-1)^2} = \sqrt{169 + 1} = \sqrt{170} \approx 13.04
   \end{align*}
   $$

6. **Distance between $$ P(3,-6) $$ and $$ Q(-7,-1) $$:**
   $$
   \begin{align*}
   \Delta x &= -7 - 3 = -10 \
   \Delta y &= -1 - (-6) = 5 \
   d &= \sqrt{(-10)^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \approx 11.18
   \end{align*}
   $$

### **Summary of Distances**

| **Pair** | **Distance (Exact Form)** | **Approximate Distance** |
|----------|---------------------------|--------------------------|
| N & O    | $$ 5\sqrt{5} $$           | 11.18 units              |
| N & P    | $$ \sqrt{170} $$          | 13.04 units              |
| N & Q    | $$ 3\sqrt{5} $$           | 6.71 units               |
| O & P    | $$ 3\sqrt{5} $$           | 6.71 units               |
| O & Q    | $$ \sqrt{170} $$          | 13.04 units              |
| P & Q    | $$ 5\sqrt{5} $$           | 11.18 units              |

### **Visualization**

To better understand the distances, you can plot the points on a coordinate plane:

- **N(-4, 5)**
- **O(6, 0)**
- **P(3, -6)**
- **Q(-7, -1)**

Connecting each pair of points will help visualize the distances calculated above.

### **Conclusion**

Using the Distance Formula, we've calculated all pairwise distances between the points $$ N $$, $$ O $$, $$ P $$, and $$ Q $$. This method is fundamental in coordinate geometry for determining the length between any two points in a plane.
The distances between the points are as follows: - $$ N(-4,5) $$ to $$ O(6,0) $$: $$ 5\sqrt{5} $$ units (approximately 11.18 units) - $$ N(-4,5) $$ to $$ P(3,-6) $$: $$ \sqrt{170} $$ units (approximately 13.04 units) - $$ N(-4,5) $$ to $$ Q(-7,-1) $$: $$ 3\sqrt{5} $$ units (approximately 6.71 units) - $$ O(6,0) $$ to $$ P(3,-6) $$: $$ 3\sqrt{5} $$ units (approximately 6.71 units) - $$ O(6,0) $$ to $$ Q(-7,-1) $$: $$ \sqrt{170} $$ units (approximately 13.04 units) - $$ P(3,-6) $$ to $$ Q(-7,-1) $$: $$ 5\sqrt{5} $$ units (approximately 11.18 units)

16. \( N(-4,5), O(6,0), P(3,-6), Q(-7,-1) \); Distance Formula

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