11. Find the distance between \( (-2,5) \) and \( (4,3) \). Keep your answer in \( \quad * \ldots / 1 \) radical form. \( \quad \vee \) (copy and paste this symbol)

To find the distance between the points \( (-2,5) \) and \( (4,3) \), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the values: \[ d = \sqrt{(4 - (-2))^2 + (3 - 5)^2} = \sqrt{(4 + 2)^2 + (-2)^2} = \sqrt{6^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} \] Thus, the distance is \( \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \). Now, isn't it fascinating how this formula not only helps calculate distances in geometry but is also deeply rooted in the Pythagorean theorem? That clever connection shows us that distances in a flat plane can be visualized as the hypotenuse of a right triangle! In real life, you can apply the concepts of distance calculations in various scenarios, such as planning travel routes, determining the closest locations (like finding a café near you), or even in fields like robotics where understanding the distance between two points is crucial for navigation and movement. It’s all about measuring that perfect path!