Question

practice and problem solving
find the coordinates of the midpoint of each segment.

1. $overline{xy}$ with endpoints $x(-3, -7)$ and $y(-1, 1)$
2. $overline{mn}$ with endpoints $m(12, -7)$ and $n(-5, -2)$
3. $m$ is the midpoint of $overline{qr}$. $q$ has coordinates $(-3, 5)$, and $m$ has coordinates $(7, -9)$. find the coordinates of $r$.
4. $d$ is the midpoint of $overline{ce}$. $e$ 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. $overline{de}$ and $overline{fg}$
2. $overline{de}$ and $overline{rs}$

Explanation:

Step1: Recall mid - point formula

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

Step2: Solve for mid - point of $\overline{XY}$

Given $X(-3,-7)$ and $Y(-1,1)$.
$x=\frac{-3+( - 1)}{2}=\frac{-3 - 1}{2}=\frac{-4}{2}=-2$
$y=\frac{-7 + 1}{2}=\frac{-6}{2}=-3$
The mid - point of $\overline{XY}$ is $(-2,-3)$.

Step3: Solve for mid - point of $\overline{MN}$

Given $M(12,-7)$ and $N(-5,-2)$.
$x=\frac{12+( - 5)}{2}=\frac{12 - 5}{2}=\frac{7}{2}=3.5$
$y=\frac{-7+( - 2)}{2}=\frac{-7 - 2}{2}=\frac{-9}{2}=-4.5$
The mid - point of $\overline{MN}$ is $(3.5,-4.5)$.

Step4: Find endpoint $R$ given mid - point $M$ and endpoint $Q$

Let $Q(-3,5)$ and $M(7,-9)$. Let the coordinates of $R$ be $(x,y)$.
Using the mid - point formula for the $x$ - coordinate: $\frac{-3 + x}{2}=7$, then $-3+x = 14$, so $x=17$.
Using the mid - point formula for the $y$ - coordinate: $\frac{5 + y}{2}=-9$, then $5 + y=-18$, so $y=-23$.
The coordinates of $R$ are $(17,-23)$.

Step5: Find endpoint $C$ given mid - point $D$ and endpoint $E$

Let $E(-3,-2)$ and $D(2.5,1)$. Let the coordinates of $C$ be $(x,y)$.
Using the mid - point formula for the $x$ - coordinate: $\frac{-3 + x}{2}=2.5$, then $-3+x = 5$, so $x = 8$.
Using the mid - point formula for the $y$ - coordinate: $\frac{-2 + y}{2}=1$, then $-2 + y=2$, so $y = 4$.
The coordinates of $C$ are $(8,4)$.

Step6: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step7: Find length of $\overline{DE}$

Let $D(-4,-4)$ and $E(0,-2)$.
$d_{DE}=\sqrt{(0-( - 4))^2+((-2)-( - 4))^2}=\sqrt{(4)^2+(2)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$

Step8: Find length of $\overline{FG}$

Let $F(2,4)$ and $G(4,0)$.
$d_{FG}=\sqrt{(4 - 2)^2+(0 - 4)^2}=\sqrt{(2)^2+( - 4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}$
Since $d_{DE}=d_{FG}$, $\overline{DE}$ and $\overline{FG}$ are congruent.

Step9: Find length of $\overline{RS}$

Let $R(-4,-4)$ and $S(2,-2)$.
$d_{RS}=\sqrt{(2-( - 4))^2+((-2)-( - 4))^2}=\sqrt{(6)^2+(2)^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10}$
Since $d_{DE}=2\sqrt{5}$ and $d_{RS}=2\sqrt{10}$, $\overline{DE}$ and $\overline{RS}$ are not congruent.

Answer:

1. $(-2,-3)$
2. $(3.5,-4.5)$
3. $(17,-23)$
4. $(8,4)$
5. $d_{DE}=2\sqrt{5}$, $d_{FG}=2\sqrt{5}$, congruent
6. $d_{DE}=2\sqrt{5}$, $d_{RS}=2\sqrt{10}$, not congruent