Question

1. given the segment with endpoints (-1,8) and (7,2), determine its midpoint and its length. 2. given the graph below, determine the segments midpoint and find its length. 3. point m is the midpoint of ab. if the coordinates of a are (-3,6) and the coordinates of m are (-5,2), what are the coordinates of b?

Explanation:

Step1: Recall mid - point and distance formulas

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 the distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Solve for problem 1

Let $(x_1,y_1)=(-1,8)$ and $(x_2,y_2)=(7,2)$.
Mid - point:
$M=(\frac{-1 + 7}{2},\frac{8+2}{2})=(\frac{6}{2},\frac{10}{2})=(3,5)$
Length:
$d=\sqrt{(7+1)^2+(2 - 8)^2}=\sqrt{8^2+( - 6)^2}=\sqrt{64 + 36}=\sqrt{100}=10$

Step3: Solve for problem 2

Let the two endpoints from the graph be $(x_1,y_1)=(0,4)$ and $(x_2,y_2)=(4,-4)$.
Mid - point:
$M=(\frac{0 + 4}{2},\frac{4-4}{2})=(2,0)$
Length:
$d=\sqrt{(4 - 0)^2+(-4 - 4)^2}=\sqrt{4^2+( - 8)^2}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}$

Step4: Solve for problem 3

Let $A(x_1,y_1)=(-3,6)$ and $M(x_m,y_m)=(-5,2)$.
Using the mid - point formula $x_m=\frac{x_1 + x_2}{2}$ and $y_m=\frac{y_1 + y_2}{2}$.
For the x - coordinate of $B$:
$-5=\frac{-3+x_2}{2}$, then $-10=-3+x_2$, so $x_2=-7$.
For the y - coordinate of $B$:
$2=\frac{6+y_2}{2}$, then $4 = 6+y_2$, so $y_2=-2$.
So $B=(-7,-2)$

Answer:

1. Mid - point: $(3,5)$, Length: $10$
2. Mid - point: $(2,0)$, Length: $4\sqrt{5}$
3. Coordinates of $B$: $(-7,-2)$