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 [tex]\( A(-6, -6) \)[/tex]
- Point [tex]\( B(-4, 1) \)[/tex]
- Point [tex]\( C(-9, 4) \)[/tex]
2. Properties of a Parallelogram:
In a parallelogram, the opposite sides are parallel and equal in length. So vector [tex]\( \overrightarrow{AB} \)[/tex] must be parallel and equal to vector [tex]\( \overrightarrow{CD} \)[/tex] and vector [tex]\( \overrightarrow{BC} \)[/tex] must be parallel and equal to vector [tex]\( \overrightarrow{AD} \)[/tex].
3. Finding Vector [tex]\( \overrightarrow{BC} \)[/tex]:
To find [tex]\( \overrightarrow{BC} \)[/tex], we subtract the coordinates of B from C:
[tex]\[
\overrightarrow{BC} = (C_1 - B_1, C_2 - B_2) = (-9 - (-4), 4 - 1) = (-5, 3)
\][/tex]
4. Using Vector [tex]\( \overrightarrow{BC} \)[/tex] to Find Point D:
Since [tex]\( \overrightarrow{AD} = \overrightarrow{BC} \)[/tex], we add [tex]\( \overrightarrow{BC} \)[/tex] to the coordinates of A:
[tex]\[
D = (A_1 + \overrightarrow{BC}_1, A_2 + \overrightarrow{BC}_2) = (-6 + (-5), -6 + 3) = (-11, -3)
\][/tex]
5. Coordinates of Point D:
Therefore, the coordinates of point D are:
[tex]\[
D(-11, -3)
\][/tex]
6. Summary:
The coordinates of point D are [tex]\((-11, -3)\)[/tex], 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.
