Question

find the coordinates of the missing endpoint if b is the midpoint of ac.

1. a(-5, 4), b(-2, 5)
2. a(1, 7), b(-3, 1)
3. a(-4, 2), b(6, -1)
4. c(-6, -2), b(-3, -5)
5. a(4, -0.25), b(-4, 6.5)
6. c(5/3, -6), b(8/3, 4)

Explanation:

Step1: Recall mid - point formula

The mid - point formula between two points $A(x_1,y_1)$ and $C(x_2,y_2)$ with mid - point $B(x_m,y_m)$ is $x_m=\frac{x_1 + x_2}{2}$ and $y_m=\frac{y_1 + y_2}{2}$. We can re - arrange these formulas to find the unknown endpoint. For the $x$ - coordinate of the unknown endpoint $x_2 = 2x_m−x_1$ and for the $y$ - coordinate $y_2 = 2y_m−y_1$.

Step2: Solve for 33

Let $A(x_1,y_1)=(-5,4)$ and $B(x_m,y_m)=(-2,5)$.
For the $x$ - coordinate of $C$: $x_2=2x_m−x_1=2\times(-2)-(-5)=-4 + 5=1$.
For the $y$ - coordinate of $C$: $y_2=2y_m−y_1=2\times5 - 4=10 - 4 = 6$. So the coordinates of $C$ are $(1,6)$.

Step3: Solve for 34

Let $A(x_1,y_1)=(1,7)$ and $B(x_m,y_m)=(-3,1)$.
For the $x$ - coordinate of $C$: $x_2=2x_m−x_1=2\times(-3)-1=-6 - 1=-7$.
For the $y$ - coordinate of $C$: $y_2=2y_m−y_1=2\times1 - 7=2 - 7=-5$. So the coordinates of $C$ are $(-7,-5)$.

Step4: Solve for 35

Let $A(x_1,y_1)=(-4,2)$ and $B(x_m,y_m)=(6,-1)$.
For the $x$ - coordinate of $C$: $x_2=2x_m−x_1=2\times6-(-4)=12 + 4 = 16$.
For the $y$ - coordinate of $C$: $y_2=2y_m−y_1=2\times(-1)-2=-2 - 2=-4$. So the coordinates of $C$ are $(16,-4)$.

Step5: Solve for 36

Let $A(x_1,y_1)=(-6,-2)$ and $B(x_m,y_m)=(-3,-5)$.
For the $x$ - coordinate of $C$: $x_2=2x_m−x_1=2\times(-3)-(-6)=-6 + 6=0$.
For the $y$ - coordinate of $C$: $y_2=2y_m−y_1=2\times(-5)-(-2)=-10 + 2=-8$. So the coordinates of $C$ are $(0,-8)$.

Step6: Solve for 37

Let $A(x_1,y_1)=(4,-0.25)$ and $B(x_m,y_m)=(-4,6.5)$.
For the $x$ - coordinate of $C$: $x_2=2x_m−x_1=2\times(-4)-4=-8 - 4=-12$.
For the $y$ - coordinate of $C$: $y_2=2y_m−y_1=2\times6.5-(-0.25)=13 + 0.25 = 13.25$. So the coordinates of $C$ are $(-12,13.25)$.

Step7: Solve for 38

Let $A(x_1,y_1)=(\frac{5}{3},-6)$ and $B(x_m,y_m)=(\frac{8}{3},4)$.
For the $x$ - coordinate of $C$: $x_2=2x_m−x_1=2\times\frac{8}{3}-\frac{5}{3}=\frac{16 - 5}{3}=\frac{11}{3}$.
For the $y$ - coordinate of $C$: $y_2=2y_m−y_1=2\times4-(-6)=8 + 6 = 14$. So the coordinates of $C$ are $(\frac{11}{3},14)$.

Answer:

1. $(1,6)$
2. $(-7,-5)$
3. $(16,-4)$
4. $(0,-8)$
5. $(-12,13.25)$
6. $(\frac{11}{3},14)$