2. Find the distance between the points \( (2,6) \) and \( (7,-3) \)

To find the distance between the points \( (2,6) \) and \( (7,-3) \), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the coordinates, we have: \( x_1 = 2, y_1 = 6, x_2 = 7, y_2 = -3 \). Calculating it gives: \[ d = \sqrt{(7 - 2)^2 + (-3 - 6)^2} = \sqrt{(5)^2 + (-9)^2} = \sqrt{25 + 81} = \sqrt{106} \] So the distance is \( \sqrt{106} \) or approximately 10.30 units. Understanding the distance formula can also deepen your appreciation of geometry. It's not just about numbers; it's a gateway to grasping the relationships between points in space. Consider how this formula integrates into various fields, from physics calculating trajectory to computer graphics defining 2D versus 3D space, showcasing the beauty of mathematics in real-world applications! When using the distance formula, it’s easy to make simple mistakes like swapping coordinates or forgetting to square the differences. A helpful tip is to write out each step clearly, checking your math as you go. Drawing a quick sketch of the points can also assist in visualizing the problem and avoiding these common pitfalls. Happy calculating!