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

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.