Question

plot the points a(-6,-4), b(4, 11), c(7, 9) on the coordinate axes below. state the coordinates of point d such that a, b, c, and d would form a rectangle. (plotting point d is optional.)
click on the graph to plot a point. click a point to delete it.

Explanation:

Step1: Recall rectangle midpoint property

In a rectangle, the diagonals share the same midpoint. For points $A(x_A,y_A)$, $B(x_B,y_B)$, $C(x_C,y_C)$, $D(x_D,y_D)$, the midpoint of $AC$ equals the midpoint of $BD$:
$$\frac{x_A+x_C}{2}=\frac{x_B+x_D}{2}, \quad \frac{y_A+y_C}{2}=\frac{y_B+y_D}{2}$$

Step2: Solve for $x_D$

Substitute $x_A=-6$, $x_C=7$, $x_B=4$ into the x-coordinate midpoint equation:
$$\frac{-6+7}{2}=\frac{4+x_D}{2}$$
Multiply both sides by 2: $1=4+x_D$, so $x_D=1-4=-3$

Step3: Solve for $y_D$

Substitute $y_A=-4$, $y_C=9$, $y_B=11$ into the y-coordinate midpoint equation:
$$\frac{-4+9}{2}=\frac{11+y_D}{2}$$
Multiply both sides by 2: $5=11+y_D$, so $y_D=5-11=-6$

Answer:

The coordinates of point $D$ are $(-3, -6)$