## Geometry: Common Core (15th Edition)

In this problem, we need to use the distance formula to find the length of $\overline{CD}$ and the midpoint formula to find its midpoint. Let's find the midpoint first. The midpoint can be found using the following formula: $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$ The points we are given are $(5, 7)$ and $(10, -5)$. Let's plug these points into the formula: $M = (\frac{5 + 10}{2}, \frac{7 + (-5)}{2})$ Use addition to simplify: $M = (\frac{15}{2}, \frac{2}{2})$ Simplify the y-coordinate by dividing the numerator and denominator by their greatest common factor, $2$: $M = (\frac{15}{2}, 1)$ The coordinates of the midpoint of $\overline{CD}$ are $(\frac{15}{2}, 1)$. The distance between two points is given by the following formula: $D = \sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Let's plug in the same points into this formula to find the distance between them: $D = \sqrt {(10 - 5)^2 + (-5 - 7)^2}$ Simplify what is inside the parentheses: $D = \sqrt {(5)^2 + (-12)^2}$ Evaluate the exponents: $D = \sqrt {25 + 144}$ Add to simplify: $D = \sqrt {169}$ Take the square root to solve for $D$: $D = 13$ $\overline{CD}$ is $13$.