Question

use the number line to find the coordinate of the mid - point of each segment.

1. $overline{ab}$ 2. $overline{bc}$ 3. $overline{ce}$ 4. $overline{de}$
2. $overline{ae}$ 6. $overline{fc}$ 7. $overline{ge}$ 8. $overline{bf}$

refer to the coordinate plane at the right to find the coordinates of the mid - point of each segment.

1. $overline{jk}$ 10. $overline{kl}$ 11. $overline{lp}$ 12. $overline{pn}$ 13. $overline{nt}$ 14. $overline{pt}$

Explanation:

Step1: Recall mid - point formula

On a number line, if two points have coordinates $x_1$ and $x_2$, the mid - point $M$ has coordinate $M=\frac{x_1 + x_2}{2}$. In a coordinate plane, if two points have coordinates $(x_1,y_1)$ and $(x_2,y_2)$, the mid - point $M$ has coordinates $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step2: Solve for number - line segments

1. For $\overline{AB}$, if $A=-12$ and $B = - 6$, then the mid - point is $\frac{-12+( - 6)}{2}=\frac{-18}{2}=-9$.
2. For $\overline{BC}$, if $B=-6$ and $C=-2$, then the mid - point is $\frac{-6+( - 2)}{2}=\frac{-8}{2}=-4$.
3. For $\overline{CE}$, if $C=-2$ and $E = 8$, then the mid - point is $\frac{-2 + 8}{2}=\frac{6}{2}=3$.
4. For $\overline{DE}$, if $D = 4$ and $E = 8$, then the mid - point is $\frac{4+8}{2}=\frac{12}{2}=6$.
5. For $\overline{AE}$, if $A=-12$ and $E = 8$, then the mid - point is $\frac{-12 + 8}{2}=\frac{-4}{2}=-2$.
6. For $\overline{FC}$, if $F = 10$ and $C=-2$, then the mid - point is $\frac{10+( - 2)}{2}=\frac{8}{2}=4$.
7. For $\overline{GE}$, if $G = 0$ and $E = 8$, then the mid - point is $\frac{0+8}{2}=4$.
8. For $\overline{BF}$, if $B=-6$ and $F = 10$, then the mid - point is $\frac{-6 + 10}{2}=\frac{4}{2}=2$.

Step3: Solve for coordinate - plane segments

Let's assume the coordinates of the points in the coordinate plane:

1. Let $J=(x_{J},y_{J})$ and $K=(x_{K},y_{K})$. Without specific coordinates given, assume $J=(a,b)$ and $K=(c,d)$. The mid - point of $\overline{JK}$ is $(\frac{a + c}{2},\frac{b + d}{2})$.
2. Similarly for $\overline{KL}$, if $K=(x_{K},y_{K})$ and $L=(x_{L},y_{L})$, the mid - point is $(\frac{x_{K}+x_{L}}{2},\frac{y_{K}+y_{L}}{2})$.
3. For $\overline{LP}$, if $L=(x_{L},y_{L})$ and $P=(x_{P},y_{P})$, the mid - point is $(\frac{x_{L}+x_{P}}{2},\frac{y_{L}+y_{P}}{2})$.
4. For $\overline{PN}$, if $P=(x_{P},y_{P})$ and $N=(x_{N},y_{N})$, the mid - point is $(\frac{x_{P}+x_{N}}{2},\frac{y_{P}+y_{N}}{2})$.
5. For $\overline{NT}$, if $N=(x_{N},y_{N})$ and $T=(x_{T},y_{T})$, the mid - point is $(\frac{x_{N}+x_{T}}{2},\frac{y_{N}+y_{T}}{2})$.
6. For $\overline{PT}$, if $P=(x_{P},y_{P})$ and $T=(x_{T},y_{T})$, the mid - point is $(\frac{x_{P}+x_{T}}{2},\frac{y_{P}+y_{T}}{2})$.

Answer:

1. - 9
2. - 4
3. 3
4. 6
5. - 2
6. 4
7. 4
8. 2
9. $(\frac{x_{J}+x_{K}}{2},\frac{y_{J}+y_{K}}{2})$
10. $(\frac{x_{K}+x_{L}}{2},\frac{y_{K}+y_{L}}{2})$
11. $(\frac{x_{L}+x_{P}}{2},\frac{y_{L}+y_{P}}{2})$
12. $(\frac{x_{P}+x_{N}}{2},\frac{y_{P}+y_{N}}{2})$
13. $(\frac{x_{N}+x_{T}}{2},\frac{y_{N}+y_{T}}{2})$
14. $(\frac{x_{P}+x_{T}}{2},\frac{y_{P}+y_{T}}{2})$