Sure, let's go through the steps to determine the coordinates of point D such that the points A, B, C, and D would form a parallelogram.
1. Plotting the Points:
Let's first plot the given points on the coordinate plane.
- Point \(\( A(-6, -6) \)\)
- Point \(\( B(-4, 1) \)\)
- Point \(\( C(-9, 4) \)\)
2. Properties of a Parallelogram:
In a parallelogram, the opposite sides are parallel and equal in length. So vector \(\( \overrightarrow{AB} \)\) must be parallel and equal to vector \(\( \overrightarrow{CD} \)\) and vector \(\( \overrightarrow{BC} \)\) must be parallel and equal to vector \(\( \overrightarrow{AD} \)\).
3. Finding Vector \(\( \overrightarrow{BC} \)\):
To find \(\( \overrightarrow{BC} \)\), we subtract the coordinates of B from C:
\(\[
\overrightarrow{BC} = (C_1 - B_1, C_2 - B_2) = (-9 - (-4), 4 - 1) = (-5, 3)
\]\)
4. Using Vector \(\( \overrightarrow{BC} \)\) to Find Point D:
Since \(\( \overrightarrow{AD} = \overrightarrow{BC} \)\), we add \(\( \overrightarrow{BC} \)\) to the coordinates of A:
\(\[
D = (A_1 + \overrightarrow{BC}_1, A_2 + \overrightarrow{BC}_2) = (-6 + (-5), -6 + 3) = (-11, -3)
\]\)
5. Coordinates of Point D:
Therefore, the coordinates of point D are:
\(\[
D(-11, -3)
\]\)
6. Summary:
The coordinates of point D are \(\((-11, -3)\)\), which make the points A, B, C, and D form a parallelogram when plotted on the coordinate axes.
Feel free to plot these points on graph paper or any graph plotter tool to visually verify that these points form a parallelogram.
