The **distance formula** is used to find the distance between two points in a coordinate plane. It's derived from the Pythagorean Theorem and works in both two-dimensional and three-dimensional spaces. 

### **Formula in Two Dimensions (2D)**

If you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \( d \) between these points is given by:

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

### **Explanation**

1. **Difference in Coordinates**: The formula calculates the horizontal distance \((x_2 - x_1)\) and the vertical distance \((y_2 - y_1)\) between the two points.
2. **Squaring**: Squaring these differences removes any negative signs and gives a measure of distance squared.
3. **Summation**: Adding the squared distances gives the total squared distance between the points.
4. **Square Root**: Finally, taking the square root of the sum converts it back into the actual distance.

### **Example Calculation**

Consider points \( A(1, 2) \) and \( B(4, 6) \). Using the formula:

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

### **Formula in Three Dimensions (3D)**

If the points have three coordinates, such as \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \), the formula is extended as:

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

Would you like more examples, details, or have any specific questions?

### Related Questions:

1. How is the distance formula derived from the Pythagorean Theorem?
2. How does the distance formula work in real-life applications?
3. How do you use the distance formula to determine if points are collinear?
4. What is the difference between the 2D and 3D distance formulas?
5. Can the distance formula be used with complex numbers?

### Tip: 
Always double-check the subtraction steps when using the distance formula to ensure that the signs are correct.

Explore the distance formula used to calculate the distance between two points on a coordinate plane. This guide covers the 2D and 3D distance formulas, with examples suitable for students in Grades 8-10, and is rooted in the Pythagorean Theorem.

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