Show thinking about how to find the distance between given points

To find the distance between two points in a coordinate system, you can use the distance formula derived from the Pythagorean theorem. Here's a step-by-step explanation of the thinking process involved:

### Steps to Find the Distance Between Two Points

1. **Identify the Points**: Let's say we have two points in a 2D coordinate plane:
- Point A: \( (x_1, y_1) \)
- Point B: \( (x_2, y_2) \)

2. **Visualize the Points**: It can help to visualize the two points on a graph. The distance you are trying to find is the length of the line segment that connects these two points.

3. **Use the Pythagorean Theorem**:
- The distance between the points can be thought of as the hypotenuse of a right triangle, where the legs of the triangle are the differences in the x-coordinates and y-coordinates of the points.
- The horizontal leg of the triangle can be calculated as the difference in the x-coordinates: \( |x_2 - x_1| \).
- The vertical leg of the triangle is the difference in the y-coordinates: \( |y_2 - y_1| \).

4. **Apply the Distance Formula**: According to the Pythagorean theorem, the square of the length of the hypotenuse (the distance \( d \)) can be found by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
This formula combines the lengths of the two legs of the triangle to calculate the distance.

5. **Calculate the Values**: To find the distance numerically, simply substitute the values of \( x_1, y_1, x_2, \) and \( y_2 \) into the distance formula and perform the calculations.

### Example

Let's consider an example:

- Point A: \( (3, 4) \)
- Point B: \( (7, 1) \)

1. Identify the coordinates:
- \( x_1 = 3, y_1 = 4 \)
- \( x_2 = 7, y_2 = 1 \)

2. Calculate the differences:
- \( |x_2 - x_1| = |7 - 3| = 4 \)
- \( |y_2 - y_1| = |1 - 4| = 3 \)

3. Apply the distance formula:
\[
d = \sqrt{(7 - 3)^2 + (1 - 4)^2}
\]
\[
d = \sqrt{(4)^2 + (-3)^2}
\]
\[
d = \sqrt{16 + 9}
\]
\[
d = \sqrt{25} = 5
\]

### Conclusion

The distance between points \( (3, 4) \) and \( (7, 1) \) is \( 5 \) units.

This method can be generalized to higher dimensions by extending the formula. For points in a 3D space, for example, the distance formula would include the z-coordinates as well.