Question

How would you set up the distance formula to find the distance between points E
𝐸
and F
𝐹

Answer 1

To find the distance between two points \( E \) and \( F \) in a Cartesian coordinate system, you can use the distance formula, which is given by:

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

Here, \( (x_1, y_1) \) are the coordinates of point \( E \), and \( (x_2, y_2) \) are the coordinates of point \( F \).

### Steps to Set Up the Distance Formula

1. Identify the coordinates of the points:
- Let point \( E \) have coordinates \( (x_1, y_1) \).
- Let point \( F \) have coordinates \( (x_2, y_2) \).

2. Substitute these coordinates into the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

3. Calculate the differences \( (x_2 - x_1) \) and \( (y_2 - y_1) \), square them, add the squares, and then take the square root of the sum to find the distance \( d \).

### Example
If \( E \) is at \( (1, 2) \) and \( F \) is at \( (4, 6) \):
- \( x_1 = 1, y_1 = 2 \)
- \( x_2 = 4, y_2 = 6 \)

Plugging these values into the formula:
\[
d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Thus, the distance between points \( E \) and \( F \) is 5 units.