Question

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

Answer 1

To find the distance between two points E and F in a Cartesian coordinate system, you use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by the following equation:

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

Where:
- \(d\) is the distance between the two points,
- \((x_1, y_1)\) are the coordinates of point E,
- \((x_2, y_2)\) are the coordinates of point F.

### Steps to use the distance formula:

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

2. **Plug the coordinates** into the distance formula:
- Substitute \(x_1\), \(y_1\), \(x_2\), and \(y_2\) into the formula.

3. **Calculate the differences** between the x-coordinates and y-coordinates:
- Compute \( (x_2 - x_1) \) and \( (y_2 - y_1) \).

4. **Square the differences**, sum them, and then take the square root to find the distance \(d\).

### Example:

If point E is at (2, 3) and point F is at (5, 7), the steps would be:

1. **Identify the coordinates**:
- \(E(2, 3)\), \(F(5, 7)\)

2. **Plug the coordinates into the distance formula**:
\[
d = \sqrt{(5 - 2)^2 + (7 - 3)^2}
\]

3. **Calculate the differences**:
- \(5 - 2 = 3\)
- \(7 - 3 = 4\)

4. **Square the differences and sum**:
\[
d = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25}
\]

5. **Calculate the square root**:
\[
d = 5
\]

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