tutor me on how to find distance between pairs of points

To find the distance between two points in a coordinate plane, you can use the **Distance Formula**. This formula is derived from the Pythagorean Theorem and works for points in both two-dimensional (2D) and three-dimensional (3D) space. Here's how you can approach it:

### 1. **Distance Between Two Points in 2D**

Consider two points, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), in a 2D plane.

**Step 1:** Identify the coordinates of both points.  
- Point A: \( (x_1, y_1) \)  
- Point B: \( (x_2, y_2) \)

**Step 2: **Subtract the coordinates to find the differences:  
- Difference in x-coordinates: \( x_2 - x_1 \)  
- Difference in y-coordinates: \( y_2 - y_1 \)

**Step 3:** Apply the **Distance Formula**:

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

Here, \( d \) represents the distance between the two points.

### Example:
Find the distance between \( A(2, 3) \) and \( B(5, 7) \).

\[
d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

So, the distance between the points is 5 units.

### 2. **Distance Between Two Points in 3D**

If you have points in 3D space, \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \), you use a similar formula with an additional term for the z-coordinates.

**Step 1:** Identify the coordinates of both points.  
- Point A: \( (x_1, y_1, z_1) \)  
- Point B: \( (x_2, y_2, z_2) \)

**Step 2:** Apply the **3D Distance Formula**:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

### Example:
Find the distance between \( A(1, 2, 3) \) and \( B(4, 6, 8) \).

\[
d = \sqrt{(4 - 1)^2 + (6 - 2)^2 + (8 - 3)^2} = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07
\]

So, the distance between the points is approximately 7.07 units.

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### Do you want more details or have any questions? 

### 5 Related Questions:
1. How do you derive the distance formula from the Pythagorean Theorem?
2. How do you find the midpoint between two points?
3. What is the distance between points on a number line?
4. How does the distance formula change if the plane is not Cartesian?
5. Can the distance formula be applied to points in higher dimensions?

### Tip:
Always double-check your calculations by squaring the differences carefully to avoid errors when finding the distance.

Learn how to calculate the distance between two points in 2D and 3D space using the Distance Formula. This tutorial covers key geometry concepts and is suitable for high school students in Grades 9-12.

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