Question

1. b is the mid - point of ac and e is the mid - point of bd. if a(-9, -4), c(-1, 6), and e(-4, -3), find the coordinates of d.

Explanation:

Step1: Find coordinates of B using mid - point formula for AC

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For points $A(-9,-4)$ and $C(-1,6)$, if $B$ is the mid - point of $AC$, then $x_B=\frac{-9+( - 1)}{2}=\frac{-9 - 1}{2}=-5$ and $y_B=\frac{-4 + 6}{2}=1$. So $B(-5,1)$.

Step2: Find coordinates of D using mid - point formula for BD

Let the coordinates of $D$ be $(x_D,y_D)$. Since $E(-4,-3)$ is the mid - point of $BD$ and $B(-5,1)$. Using the mid - point formula: $\frac{-5+x_D}{2}=-4$ and $\frac{1 + y_D}{2}=-3$.
First, solve $\frac{-5+x_D}{2}=-4$:
Multiply both sides by 2: $-5+x_D=-8$. Then $x_D=-8 + 5=-3$.
Second, solve $\frac{1 + y_D}{2}=-3$:
Multiply both sides by 2: $1 + y_D=-6$. Then $y_D=-6 - 1=-7$.

Answer:

$D(-3,-7)$