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:

Step1: Recall mid - point formula

The mid - point formula between 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: Recall distance formula

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

Question 9
a)
Given $A(4,7)$ and $B(14, - 3)$
The mid - point of $\overline{AB}$ is $(\frac{4 + 14}{2},\frac{7+( - 3)}{2})=(\frac{18}{2},\frac{4}{2})=(9,2)$

b)
Let $A(4,7)$ and $B(14, - 3)$ and $B$ be the mid - point of $\overline{AE}$. Let $E=(x,y)$.
Using the mid - point formula: $\frac{4 + x}{2}=14$ and $\frac{7 + y}{2}=-3$
For $\frac{4 + x}{2}=14$, we have $4+x = 28$, so $x=24$.
For $\frac{7 + y}{2}=-3$, we have $7 + y=-6$, so $y=-13$. So $E=(24,-13)$

c)
Using the distance formula with $A(4,7)$ and $B(14, - 3)$
$d=\sqrt{(14 - 4)^2+( - 3 - 7)^2}=\sqrt{10^2+( - 10)^2}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}$

Question 10
a)
Given $A(3,-2)$ and $B(17,-11)$
The mid - point of $\overline{AB}$ is $(\frac{3+17}{2},\frac{-2+( - 11)}{2})=(\frac{20}{2},\frac{-13}{2})=(10,-\frac{13}{2})$

b)
Let $A(3,-2)$ and $B(17,-11)$ and $B$ be the mid - point of $\overline{AE}$. Let $E=(x,y)$.
Using the mid - point formula: $\frac{3 + x}{2}=17$ and $\frac{-2 + y}{2}=-11$
For $\frac{3 + x}{2}=17$, we have $3+x = 34$, so $x = 31$.
For $\frac{-2 + y}{2}=-11$, we have $-2 + y=-22$, so $y=-20$. So $E=(31,-20)$

c)
Using the distance formula with $A(3,-2)$ and $B(17,-11)$
$d=\sqrt{(17 - 3)^2+( - 11+2)^2}=\sqrt{14^2+( - 9)^2}=\sqrt{196 + 81}=\sqrt{277}$

Extra Credit
Since $N$ is the mid - point of $\overline{BM}$ and $BN = 4$, then $BM=2BN = 8$.
Since $M$ is the mid - point of $\overline{AB}$, then $AB = 2BM$. So $AB=16$

Answer:

Question 9
a) $(9,2)$
b) $(24,-13)$
c) $10\sqrt{2}$
Question 10
a) $(10,-\frac{13}{2})$
b) $(31,-20)$
c) $\sqrt{277}$
Extra Credit
$16$