Question

homework: find the midpoint of the following sets of points.

1. (-4,4) &(-2,2)
2. (2,6) & (8,-4)
3. (-7,7) & (-10,-5)

date:
find the missing coordinate given an endpoint a and midpoint m.

1. a(-1, 9) m(-9,-10)
2. a(2,5) m(5,1)
3. a(-6,4) m(4,8)

Explanation:

Step1: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. And if we know an endpoint $A(x_1,y_1)$ and mid - point $M(x_m,y_m)$, we can find the other endpoint $B(x_2,y_2)$ using $x_m=\frac{x_1 + x_2}{2}$ and $y_m=\frac{y_1 + y_2}{2}$, which gives $x_2 = 2x_m-x_1$ and $y_2 = 2y_m-y_1$.

Step2: Find mid - points

For $(-4,4)$ and $(-2,2)$

$x=\frac{-4+( - 2)}{2}=\frac{-4 - 2}{2}=\frac{-6}{2}=-3$
$y=\frac{4 + 2}{2}=\frac{6}{2}=3$
The mid - point is $(-3,3)$.

For $(2,6)$ and $(8,-4)$

$x=\frac{2 + 8}{2}=\frac{10}{2}=5$
$y=\frac{6+( - 4)}{2}=\frac{6 - 4}{2}=1$
The mid - point is $(5,1)$.

For $(-7,7)$ and $(-10,-5)$

$x=\frac{-7+( - 10)}{2}=\frac{-7 - 10}{2}=\frac{-17}{2}=-8.5$
$y=\frac{7+( - 5)}{2}=\frac{7 - 5}{2}=1$
The mid - point is $(-8.5,1)$.

Step3: Find missing coordinates

For $A(-1,9)$ and $M(-9,-10)$

$x_2=2\times(-9)-(-1)=-18 + 1=-17$
$y_2=2\times(-10)-9=-20 - 9=-29$
The missing endpoint is $(-17,-29)$.

For $A(2,5)$ and $M(5,1)$

$x_2=2\times5 - 2=10 - 2=8$
$y_2=2\times1 - 5=2 - 5=-3$
The missing endpoint is $(8,-3)$.

For $A(-6,4)$ and $M(4,8)$

$x_2=2\times4-(-6)=8 + 6=14$
$y_2=2\times8 - 4=16 - 4=12$
The missing endpoint is $(14,12)$.

Answer:

1. Mid - point of $(-4,4)$ and $(-2,2)$: $(-3,3)$
2. Mid - point of $(2,6)$ and $(8,-4)$: $(5,1)$
3. Mid - point of $(-7,7)$ and $(-10,-5)$: $(-8.5,1)$
4. Missing endpoint for $A(-1,9)$ and $M(-9,-10)$: $(-17,-29)$
5. Missing endpoint for $A(2,5)$ and $M(5,1)$: $(8,-3)$
6. Missing endpoint for $A(-6,4)$ and $M(4,8)$: $(14,12)$