We are given a parallelogram $ABCD$ with coordinates $A(5, 6)$, $B(6, 8)$, and $C(12, 11)$. We are asked to find the coordinates of point $D$.

1. Problem Description

We are given a parallelogram $ABCD$ with coordinates $A(5, 6)$, $B(6, 8)$, and $C(12, 11)$. We are asked to find the coordinates of point $D$.

2. Solution Steps

In a parallelogram, opposite sides are parallel and equal in length. This means that $\vec{AB} = \vec{DC}$.

Let the coordinates of $D$ be $(x, y)$.

Then, $\vec{AB} = (6-5, 8-6) = (1, 2)$.

And, $\vec{DC} = (12-x, 11-y)$.

Since $\vec{AB} = \vec{DC}$, we have:

$1 = 12 - x$ and $2 = 11 - y$.

Solving for $x$:

$x = 12 - 1 = 11$.

Solving for $y$:

$y = 11 - 2 = 9$.

Therefore, the coordinates of point $D$ are $(11, 9)$.

Alternatively, we can use the property that the diagonals of a parallelogram bisect each other. Let $M$ be the midpoint of $AC$ and $N$ be the midpoint of $BD$. Since $ABCD$ is a parallelogram, $M$ and $N$ are the same point.

The midpoint $M$ of $AC$ is given by $M = (\frac{5+12}{2}, \frac{6+11}{2}) = (\frac{17}{2}, \frac{17}{2})$.

The midpoint $N$ of $BD$ is given by $N = (\frac{6+x}{2}, \frac{8+y}{2})$.

Since $M = N$, we have:

$\frac{17}{2} = \frac{6+x}{2}$ and $\frac{17}{2} = \frac{8+y}{2}$.

Solving for $x$:

$17 = 6 + x \implies x = 17 - 6 = 11$.

Solving for $y$:

$17 = 8 + y \implies y = 17 - 8 = 9$.

Therefore, the coordinates of point $D$ are $(11, 9)$.

3. Final Answer

The coordinates of point D are (11, 9).