Find the distance between the points \( (0, 0) \) and \( (-5, 12) \).

To find the distance between the points \( (0, 0) \) and \( (-5, 12) \), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the coordinates \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (-5, 12) \): \[ d = \sqrt{((-5) - 0)^2 + (12 - 0)^2} = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] So, the distance between the points \( (0, 0) \) and \( (-5, 12) \) is \( 13 \). Did you know that the distance formula is rooted in the Pythagorean theorem? This theorem, named after the ancient Greek mathematician Pythagoras, was used long before the formula was formally derived, showcasing how geometry evolves with mathematics! Real-world applications of calculating distance using these concepts can be found everywhere. For example, GPS technology relies on distance calculations to determine the shortest routes between two locations. So, next time you’re navigating your way through a new city, remember that some ancient math is keeping you on the right track!