Question

1. given the two points a (4, 7) and b (14, -3). find the following:

a) find the mid - point of $overline{ab}$.
b) find a third point e such that b is the mid - point of $overline{ae}$.
c) find the length of ab.

1. given the two points a (3, -2) and b (17, -11). find the following:

a) find the mid - point of $overline{ab}$.
b) find a third point e such that b is the mid - point of $overline{ae}$.
c) find the length of ab.
extra credit (worth 10%):
m is the mid - point of $overline{ab}$ and n is the mid - point of $overline{bm}$. if bn = 4, then what is the length of ab? (hint: draw a picture.)

Explanation:

9.

a)

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 $x_1 = 4,y_1=7,x_2 = 14,y_2=-3$.
$(\frac{4 + 14}{2},\frac{7+( - 3)}{2})=(\frac{18}{2},\frac{4}{2})=(9,2)$

Step1: Let the coordinates of point $E$ be $(x,y)$. Since $B(14,-3)$ is the mid - point of $A(4,7)$ and $E(x,y)$, use the mid - point formula.

$\frac{4 + x}{2}=14$ and $\frac{7 + y}{2}=-3$.

Step2: Solve for $x$

$4+x = 28$, so $x=24$.

Step3: Solve for $y$

$7 + y=-6$, so $y=-13$.

Step1: Recall the distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here $x_1 = 4,y_1 = 7,x_2=14,y_2=-3$.
$d=\sqrt{(14 - 4)^2+(-3 - 7)^2}=\sqrt{10^2+( - 10)^2}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}$

Answer:

$(9,2)$

b)