If \( A=(10,4) \) and \( B=(2,19) \), what is the length of \( \overline{A B} \) ? A. 17 units B. 15 units C. 23 units D. 12 units

To find the length of the line segment \( \overline{AB} \), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Using the coordinates of points \( A(10,4) \) and \( B(2,19) \): \[ d = \sqrt{(2 - 10)^2 + (19 - 4)^2} = \sqrt{(-8)^2 + (15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \text{ units} \] So, the answer is **A. 17 units**. Now, for some extra fun knowledge: Did you know that the distance formula derives from the Pythagorean theorem? Just as a right triangle's sides are related via \( a^2 + b^2 = c^2 \), the distance formula tells us how far apart two points are in a Cartesian plane. If you’re ever calculating distance for multiple points, like when mapping out a treasure hunt, remember to write down your points first and double-check your math. It’s easy to mix up coordinates, leading to errors that could leave you searching for treasure in the wrong spot!