Question

plot the points a(5,-3), b(-4, 2), c(10, 6) 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. answer attempt 1 out of 10 d = (\square, \square) submit answer

Explanation:

Step1: Recall rectangle midpoint property

In a rectangle, 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 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 $x$-coordinate of $D$

Substitute $x_A=5$, $x_C=10$, $x_B=-4$:
$$\frac{5+10}{2}=\frac{-4+x_D}{2}$$
Multiply both sides by 2: $15=-4+x_D$, so $x_D=15+4=19$

Step3: Solve for $y$-coordinate of $D$

Substitute $y_A=-3$, $y_C=6$, $y_B=2$:
$$\frac{-3+6}{2}=\frac{2+y_D}{2}$$
Multiply both sides by 2: $3=2+y_D$, so $y_D=3-2=1$

Answer:

$D=(19, 1)$