Question

1. rectangle wxyz has consecutive vertices x(6, -3), y(6, -10), and z(3, -10). find the coordinates of vertex w. (example 1)

Explanation:

Step1: Identify rectangle side properties

In a rectangle, opposite sides are equal and parallel. So, $\overrightarrow{WX} = \overrightarrow{ZY}$ and $\overrightarrow{WZ} = \overrightarrow{XY}$.

Step2: Calculate vector $\overrightarrow{ZY}$

$Z(3,-10)$, $Y(6,-10)$: $\overrightarrow{ZY} = (6-3, -10-(-10)) = (3,0)$

Step3: Use $\overrightarrow{WX}=\overrightarrow{ZY}$ to find W

Let $W=(x,y)$, $X(6,-3)$. $\overrightarrow{WX}=(6-x, -3-y)=(3,0)$
Solve $6-x=3 \implies x=3$; $-3-y=0 \implies y=-3$
Verify with $\overrightarrow{XY}$: $\overrightarrow{XY}=(6-6, -10-(-3))=(0,-7)$
$\overrightarrow{WZ}=(3-x, -10-y)=(3-3, -10-(-3))=(0,-7)=\overrightarrow{XY}$, which is correct.

Answer:

$(3, -3)$