Question

How do I calculate the distance between two points using the Pythagorean Theorem?

Answer 1

To calculate the distance between two points using the Pythagorean theorem, follow these steps:

1. **Identify the two points**: Let's denote the two points as \( A(x_1, y_1) \) and \( B(x_2, y_2) \).

2. **Find the differences in coordinates**:
- Calculate the difference in the x-coordinates: \( \Delta x = x_2 - x_1 \)
- Calculate the difference in the y-coordinates: \( \Delta y = y_2 - y_1 \)

3. **Apply the Pythagorean theorem**: The distance \( d \) between the two points can be calculated using the formula:
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
This formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the distance between the two points, in this case) is equal to the sum of the squares of the lengths of the other two sides (which are \( \Delta x \) and \( \Delta y \)).

4. **Calculate the distance**: Substitute the values of \( \Delta x \) and \( \Delta y \) into the distance formula and compute the result.

### Example
Suppose you have two points \( A(3, 4) \) and \( B(7, 1) \).

1. Calculate the differences:
\[
\Delta x = 7 - 3 = 4
\]
\[
\Delta y = 1 - 4 = -3
\]

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

Thus, the distance between points A and B is 5 units.