Coordinates on Parallelograms

• Vector Translation Principle: Since opposite sides are parallel and equal, the horizontal and vertical displacement from one vertex to another must be identical for the opposite pair. For example, the vector $\vec{AB}$ is equal to the vector $\vec{DC}$.

• Diagonal Bisection Principle: The midpoint of diagonal $AC$ is mathematically identical to the midpoint of diagonal $BD$. This creates a system of equations that can be solved for any missing coordinate.

• Midpoint Formula: The coordinates of the midpoint between $(x_1, y_1)$ and $(x_2, y_2)$ are calculated as $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

Method 1: Vector Translation (Counting Squares)

• Step 1: Identify a pair of vertices where both coordinates are known (e.g., $A$ and $B$).
• Step 2: Calculate the change in $x$ ($\Delta x = x_B - x_A$) and the change in $y$ ($\Delta y = y_B - y_A$).
• Step 3: Apply this same change to the third known vertex (e.g., $D$) to find the fourth vertex ($C$): $x_C = x_D + \Delta x$ and $y_C = y_D + \Delta y$.

Method 2: Midpoint Equality

• Step 1: Identify the two diagonals (e.g., $AC$ and $BD$).
• Step 2: Calculate the midpoint of the diagonal where both endpoints are known using the midpoint formula.
• Step 3: Set the midpoint of the second diagonal equal to the result from Step 2 and solve for the unknown $x$ and $y$ values.

• Use the Vector Method when you want a quick visual or arithmetic solution without fractions.
• Use the Midpoint Method when you need to verify if a shape is a parallelogram or when working with algebraic expressions.

• Sketch First: Always draw a rough sketch of the points on a set of axes to ensure the vertex order (e.g., $ABCD$) makes sense; the points must follow the perimeter in order.

• Check the Labeling: If the question says parallelogram $PQRS$, the diagonals are $PR$ and $QS$. Connecting $PQ$ and $RS$ as diagonals is a common error that leads to the wrong vertex.

• Verify with Slopes: After finding a missing vertex, you can verify your answer by checking that the slopes of opposite sides are equal ($m = \frac{y_2 - y_1}{x_2 - x_1}$).

• Sanity Check: Ensure the calculated point 'looks' right on your sketch; if your point is far away from where the fourth corner should be, re-check your arithmetic.

• Incorrect Diagonal Pairing: Students often assume $AB$ and $CD$ are diagonals. In a parallelogram $ABCD$, the diagonals are always the non-adjacent vertices: $AC$ and $BD$.

• Sign Errors: When calculating $\Delta x$ or $\Delta y$, subtracting a negative number (e.g., $5 - (-2) = 7$) is a frequent source of mistakes.

• Midpoint Confusion: Forgetting to divide by 2 in the midpoint formula or accidentally subtracting the coordinates instead of adding them.