Question

practice and problem solving
find the coordinates of the mid - point of each segment.

1. $xy$ with endpoints $x(-3,-7)$ and $y(-1,1)$
2. $mn$ with endpoints $m(12,-7)$ and $n(-5,-2)$
3. $m$ is the mid - point of $qr$. $q$ has coordinates $(-3,5)$, and $m$ has coordinates $(7,-9)$. find the coordinates of $r$.
4. $d$ is the mid - point of $ce$. $e$ has coordinates $(-3,-2)$, and $d$ has coordinates $(2\frac{1}{2},1)$. find the coordinates of $c$.

multi - step find the length of the given segments and determine if they are congruent.

1. $de$ and $fg$
2. $de$ and $rs$

1 - 6 midpoint and distance in the coordinate plane 47

Explanation:

12.

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1=-3,y_1 = - 7,x_2=-1,y_2 = 1$.

Step2: Calculate x - coordinate of mid - point

$x=\frac{-3+( - 1)}{2}=\frac{-3 - 1}{2}=\frac{-4}{2}=-2$.

Step3: Calculate y - coordinate of mid - point

$y=\frac{-7 + 1}{2}=\frac{-6}{2}=-3$.

Step1: Apply mid - point formula

For points $M(12,-7)$ and $N(-5,-2)$ with the mid - point formula $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$, where $x_1 = 12,y_1=-7,x_2=-5,y_2=-2$.

Step2: Find x - coordinate of mid - point

$x=\frac{12+( - 5)}{2}=\frac{12 - 5}{2}=\frac{7}{2}=3.5$.

Step3: Find y - coordinate of mid - point

$y=\frac{-7+( - 2)}{2}=\frac{-7 - 2}{2}=\frac{-9}{2}=-4.5$.

Step1: Let the coordinates of $R$ be $(x,y)$.

Since $M$ is the mid - point of $QR$, using the mid - point formula $(\frac{x_Q+x_R}{2},\frac{y_Q + y_R}{2})=(x_M,y_M)$. We know $x_Q=-3,y_Q = 5,x_M = 7,y_M=-9$.

Step2: Solve for x - coordinate of $R$

$\frac{-3+x}{2}=7$. Cross - multiply: $-3+x = 14$. Then $x=14 + 3=17$.

Step3: Solve for y - coordinate of $R$

$\frac{5+y}{2}=-9$. Cross - multiply: $5+y=-18$. Then $y=-18 - 5=-23$.

Answer:

$(-2,-3)$

13.