Question

1. m is the mid - point of ab, find the coordinates of a if m(-3, 1) and b(2, 0)
2. m is the mid - point of dc, find the coordinates of a if m(-4, 2) and c(6, 2)
3. given directed line segment ab below, find the coordinates of c on ab such that the ratio

Explanation:

Step1: Recall mid - point formula

The mid - point formula for 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: Solve for coordinates of A in first case

Let $A(x,y)$, $M(-3,1)$ and $B(2,0)$. Then $\frac{x + 2}{2}=-3$ and $\frac{y+0}{2}=1$. Solving $\frac{x + 2}{2}=-3$ gives $x+2=-6$, so $x=-8$. Solving $\frac{y}{2}=1$ gives $y = 2$. So $A(-8,2)$.

Step3: Solve for coordinates of A in second case

Let $A(x,y)$, $M(-4,2)$ and $C(6,2)$. Then $\frac{x + 6}{2}=-4$ and $\frac{y + 2}{2}=2$. Solving $\frac{x+6}{2}=-4$ gives $x+6=-8$, so $x=-14$. Solving $\frac{y + 2}{2}=2$ gives $y+2 = 4$, so $y=2$. So $A(-14,2)$.

Answer:

For first problem: $A(-8,2)$
For second problem: $A(-14,2)$