Math Problem Solver

\(4 \sqrt{10} \approx 12.6491106406735\) is the distance between \(\left(-8, 2\right)\) and \(\left(4, -2\right)\).

The distance between two points in the Cartesian plane is defined as the length of the line segment connecting the points. The formula for the distance between two points is derived from the Pythagorean Theorem using a right triangle constructed with the given points as vertices and the distance between them as the hypotenuse. The distance between \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is given by the formula: \[ \text{distance} = \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2} \] Therefore: \[ \begin{align*} \text{distance} &= \sqrt{(-8 - 4)^2 + (2 - -2)^2}\\ &=\sqrt{ \left(-12\right)^{2} + 4^{2} }\\ &=\sqrt{ 144 + 16 }\\ &=4 \sqrt{10} \end{align*} \]