Question

1. gabby is designing a quadrilateral on the coordinate plane. she knows that three of the vertices are a(0,0), b(5,2), and c (6,5). she is not sure where to graph point d. if shed like to design a parallelogram, what are the coordinates of the point d? a) (1,3) b) (3,3) c) (3,1) d) (1,5)

Explanation:

Step1: Recall parallelogram vector rule

In a parallelogram, $\vec{AB} = \vec{DC}$. For points $A(x_A,y_A)$, $B(x_B,y_B)$, $C(x_C,y_C)$, $D(x_D,y_D)$, this means $x_B - x_A = x_C - x_D$ and $y_B - y_A = y_C - y_D$.

Step2: Plug in known coordinates

$A(0,0)$, $B(5,2)$, $C(6,5)$.
For x-coordinate: $5 - 0 = 6 - x_D$
For y-coordinate: $2 - 0 = 5 - y_D$

Step3: Solve for $x_D$ and $y_D$

Solve x: $5 = 6 - x_D \implies x_D = 6 - 5 = 1$
Solve y: $2 = 5 - y_D \implies y_D = 5 - 2 = 3$

Answer:

a) (1,3)