To find the distance between two points \( J(-3, 2) \) and \( K(1, -6) \) in a 2D coordinate plane, we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) \) are the coordinates of point \( J \) and \( (x_2, y_2) \) are the coordinates of point \( K \).
Plugging in the values:
- \( x_1 = -3 \)
- \( y_1 = 2 \)
- \( x_2 = 1 \)
- \( y_2 = -6 \)
Now we calculate:
- Calculate \( x_2 - x_1 \):
\[ x_2 - x_1 = 1 - (-3) = 1 + 3 = 4 \]
- Calculate \( y_2 - y_1 \):
\[ y_2 - y_1 = -6 - 2 = -8 \]
- Apply the distance formula:
\[ d = \sqrt{(4)^2 + (-8)^2} \]
Calculating the squares:
\[ d = \sqrt{16 + 64} = \sqrt{80} \]
Simplifying \( \sqrt{80} \):
\[ d = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5} \]
Thus, the distance between points \( J \) and \( K \) is:
\[ \boxed{4\sqrt{5}} \]