Find the distance between the two points. \[ (x, y)=(-17,-2) \text { and }(x, y)=(-1,7) \] \( \square \) Additional Materials

To find the distance between the points \((-17,-2)\) and \((-1,7)\), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting in the coordinates, we have: \[ d = \sqrt{((-1) - (-17))^2 + (7 - (-2))^2} \] \[ d = \sqrt{(16)^2 + (9)^2} \] \[ d = \sqrt{256 + 81} \] \[ d = \sqrt{337} \approx 18.36 \] So, the distance between the two points is approximately 18.36 units. --- Did you know that the distance formula is fundamentally derived from the Pythagorean theorem? It essentially treats the straight line connecting two points as the hypotenuse of a right triangle, with the differences in the x-coordinates and y-coordinates forming the other two sides. This fascinating connection illustrates how geometry can help quantify our world! In practical scenarios, this distance formula can be incredibly useful! Whether you’re planning the shortest route for a road trip, determining the distance between two cities on a map, or even measuring how far you are from a friend’s house, this formula simplifies otherwise complex navigation. Plus, it’s a key principle in fields like computer graphics and physics, making it a handy tool in your mathematical toolbox!