Question

calculate the distance and the mid - point between the following points.

1. a(2, - 9) and b(-1,4)
2. c(5,9) and d(-7,-7)
3. e(8,5) and f(-1,3)
4. g(-10,-7) and h(-8,1)
5. i(-6,-10) and j(-2,-10)
6. k(3,8) and l(9,10)

Explanation:

Step1: Recall distance formula

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

Step2: For points $A(2,-9)$ and $B(-1,4)$

Distance:
\[

$$\begin{align*} d&=\sqrt{(-1 - 2)^2+(4+9)^2}\\ &=\sqrt{(-3)^2+(13)^2}\\ &=\sqrt{9 + 169}\\ &=\sqrt{178}\approx 13.34 \end{align*}$$

\]
Mid - point: $M=(\frac{2+( - 1)}{2},\frac{-9 + 4}{2})=(\frac{1}{2},-\frac{5}{2})$

Step3: For points $C(5,9)$ and $D(-7,-7)$

Distance:
\[

$$\begin{align*} d&=\sqrt{(-7 - 5)^2+(-7 - 9)^2}\\ &=\sqrt{(-12)^2+(-16)^2}\\ &=\sqrt{144 + 256}\\ &=\sqrt{400}=20 \end{align*}$$

\]
Mid - point: $M=(\frac{5+( - 7)}{2},\frac{9+( - 7)}{2})=(-1,1)$

Step4: For points $E(8,5)$ and $F(-1,3)$

Distance:
\[

$$\begin{align*} d&=\sqrt{(-1 - 8)^2+(3 - 5)^2}\\ &=\sqrt{(-9)^2+(-2)^2}\\ &=\sqrt{81 + 4}\\ &=\sqrt{85}\approx9.22 \end{align*}$$

\]
Mid - point: $M=(\frac{8+( - 1)}{2},\frac{5 + 3}{2})=(\frac{7}{2},4)$

Step5: For points $G(-10,-7)$ and $H(-8,1)$

Distance:
\[

$$\begin{align*} d&=\sqrt{(-8 + 10)^2+(1 + 7)^2}\\ &=\sqrt{(2)^2+(8)^2}\\ &=\sqrt{4+64}\\ &=\sqrt{68}\approx 8.25 \end{align*}$$

\]
Mid - point: $M=(\frac{-10+( - 8)}{2},\frac{-7 + 1}{2})=(-9,-3)$

Step6: For points $I(-6,-10)$ and $J(-2,-10)$

Distance:
\[

$$\begin{align*} d&=\sqrt{(-2 + 6)^2+(-10 + 10)^2}\\ &=\sqrt{(4)^2+(0)^2}\\ &=\sqrt{16}=4 \end{align*}$$

\]
Mid - point: $M=(\frac{-6+( - 2)}{2},\frac{-10+( - 10)}{2})=(-4,-10)$

Step7: For points $K(3,8)$ and $L(9,10)$

Distance:
\[

$$\begin{align*} d&=\sqrt{(9 - 3)^2+(10 - 8)^2}\\ &=\sqrt{(6)^2+(2)^2}\\ &=\sqrt{36 + 4}\\ &=\sqrt{40}\approx6.32 \end{align*}$$

\]
Mid - point: $M=(\frac{3 + 9}{2},\frac{8 + 10}{2})=(6,9)$

Answer:

1. Distance: $\sqrt{178}\approx 13.34$, Mid - point: $(\frac{1}{2},-\frac{5}{2})$
2. Distance: $20$, Mid - point: $(-1,1)$
3. Distance: $\sqrt{85}\approx9.22$, Mid - point: $(\frac{7}{2},4)$
4. Distance: $\sqrt{68}\approx 8.25$, Mid - point: $(-9,-3)$
5. Distance: $4$, Mid - point: $(-4,-10)$
6. Distance: $\sqrt{40}\approx6.32$, Mid - point: $(6,9)$