Find the distance between the points (3, 2) and (-1, -2).

Steps to Solve

1. Recall the distance formula

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a 2-dimensional space is given by the distance formula:

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

2. Identify the coordinates

We are given the points (3, 2) and (-1, -2). Let's assign the coordinates: $x_1 = 3$, $y_1 = 2$, $x_2 = -1$, and $y_2 = -2$.

3. Substitute the coordinates into the distance formula

Substitute the values into the distance formula:

$$ d = \sqrt{(-1 - 3)^2 + (-2 - 2)^2} $$

4. Simplify the expression

Simplify the expression inside the square root:

$$ d = \sqrt{(-4)^2 + (-4)^2} $$ $$ d = \sqrt{16 + 16} $$ $$ d = \sqrt{32} $$

5. Simplify the square root

Simplify the square root. Since $32 = 16 \times 2$, we have $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$.

$$ d = 4\sqrt{2} $$