Question

a line segment is drawn on a coordinate plane with the endpoints m(5, 9) and p(-7, -7). point q is the midpoint of $overline{mp}$. what is the length of $overline{mp}$? what is the length of $overline{qp}$? what are the coordinates of point q?

Explanation:

Step1: Calculate length of $\overline{MP}$ using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1 = 5,y_1=9,x_2=-7,y_2 = - 7$.
$d_{MP}=\sqrt{(-7 - 5)^2+(-7 - 9)^2}=\sqrt{(-12)^2+(-16)^2}=\sqrt{144 + 256}=\sqrt{400}=20$

Step2: Calculate length of $\overline{QP}$

Since $Q$ is the mid - point of $\overline{MP}$, $QP=\frac{1}{2}MP$. So $d_{QP}=\frac{20}{2}=10$

Step3: Calculate coordinates of point $Q$

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.
$x_Q=\frac{5+( - 7)}{2}=\frac{5 - 7}{2}=-1$
$y_Q=\frac{9+( - 7)}{2}=\frac{9 - 7}{2}=1$
So the coordinates of $Q$ are $(-1,1)$

Answer:

Length of $\overline{MP}$: 20
Length of $\overline{QP}$: 10
Coordinates of point $Q$: $(-1,1)$