Parallelogram Coordinates Calculation

To find the coordinates of vertex S in parallelogram PQRS, we can use the property that opposite sides are parallel and equal in length, meaning the vectors representing opposite sides are equal.

Let the coordinates be P = $(-1, -1)$, Q = $(8, 0)$, R = $(7, 5)$, and S = $(x, y)$.

In a parallelogram PQRS, the vector $\vec{PQ}$ is equal to the vector $\vec{SR}$.

Vector Calculation

Calculate the vector $\vec{PQ}$:

$\vec{PQ} = Q - P = (8 - (-1), 0 - (-1)) = (8 + 1, 0 + 1) = (9, 1)$

Calculate the vector $\vec{SR}$:

$\vec{SR} = R - S = (7 - x, 5 - y)$

Equating Vectors

Since $\vec{PQ} = \vec{SR}$, we equate their components:

• For the x-component: $9 = 7 - x$
• For the y-component: $1 = 5 - y$

Solving for S Coordinates

Solve the equations:

From $9 = 7 - x$, we get $x = 7 - 9 = -2$.

From $1 = 5 - y$, we get $y = 5 - 1 = 4$.

Therefore, the coordinates of S are $(-2, 4)$.