Step 1: Conceptual Foundation:
The distance between two points in a Cartesian coordinate system is determined using the distance formula, which is an application of the Pythagorean theorem.
Step 2: Core Formula:
The distance \(d\) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as follows:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Calculation Process:
Given point A with coordinates \( (x_1, y_1) = (-5, 7) \).
Given point B with coordinates \( (x_2, y_2) = (-1, 3) \).
Substitute these coordinates into the distance formula:
\[ d = \sqrt{(-1 - (-5))^2 + (3 - 7)^2} \]
Simplify the expression, paying attention to the double negative:
\[ d = \sqrt{(-1 + 5)^2 + (-4)^2} \]
\[ d = \sqrt{(4)^2 + (-4)^2} \]
\[ d = \sqrt{16 + 16} \]
\[ d = \sqrt{32} \]
To simplify \( \sqrt{32} \), identify its largest perfect square factor, which is 16 (\(32 = 16 \times 2\)).
\[ d = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]
Step 4: Conclusive Result:
The calculated distance between point A and point B is \(4\sqrt{2}\) units.