Question

directions: find the missing endpoint if s is the midpoint $overline{rt}$. 10. $r(-9,4)$ and $s(2, - 1)$: find $t$. 11. $s(-4,-6)$ and $t(-7,-3)$: find $r$. 12. $b$ is the midpoint of $overline{ac}$ and $e$ is the midpoint of $overline{bd}$. if $a(-9,-4),c(-1,6)$, and $e(-4,-3)$, find the coordinates of $d$. directions: suppose $q$ is the midpoint of $overline{pr}$. use the information to find the missing value. 13. $pq = 3x + 14$ and $qr = 7x - 10$: find $x$. 14. $pq = 2x + 1$ and $qr = 5x - 44$: find $pq$. 15. $pq = 6x + 25$ and $qr = 16 - 3x$: find $pr$. 16. $pr = 9x - 31$ and $qr = 43$: find $x$.

Explanation:

10.

Step1: Use mid - point formula for x - coordinate

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let $R(-9,4)$ be $(x_1,y_1)$ and $T(x,y)$ be $(x_2,y_2)$ and $S(2,-1)$ be the mid - point. For the x - coordinate, $\frac{-9 + x}{2}=2$. Cross - multiply: $-9 + x=4$. Then $x=4 + 9=13$.

Step2: Use mid - point formula for y - coordinate

For the y - coordinate, $\frac{4 + y}{2}=-1$. Cross - multiply: $4 + y=-2$. Then $y=-2 - 4=-6$.

Step1: Use mid - point formula for x - coordinate

Let $R(x,y)$ be $(x_1,y_1)$ and $T(-7,-3)$ be $(x_2,y_2)$ and $S(-4,-6)$ be the mid - point. For the x - coordinate, $\frac{x+( - 7)}{2}=-4$. Cross - multiply: $x-7=-8$. Then $x=-8 + 7=-1$.

Step2: Use mid - point formula for y - coordinate

For the y - coordinate, $\frac{y+( - 3)}{2}=-6$. Cross - multiply: $y-3=-12$. Then $y=-12 + 3=-9$.

Step1: Find the coordinates of B

First, find the mid - point $B$ of $A(-9,-4)$ and $C(-1,6)$ using the mid - point formula $B=(\frac{-9+( - 1)}{2},\frac{-4 + 6}{2})=(\frac{-10}{2},\frac{2}{2})=(-5,1)$.

Step2: Find the coordinates of D

Let $B(-5,1)$ be $(x_1,y_1)$ and $D(x,y)$ be $(x_2,y_2)$ and $E(-4,-3)$ be the mid - point. For the x - coordinate, $\frac{-5 + x}{2}=-4$. Cross - multiply: $-5 + x=-8$. Then $x=-8 + 5=-3$. For the y - coordinate, $\frac{1 + y}{2}=-3$. Cross - multiply: $1 + y=-6$. Then $y=-6 - 1=-7$.

Answer:

$T(13,-6)$

11.