Answer
13
Work Step by Step
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 $13$.
