Let's solve the problem of finding the coordinates of the fourth vertex [tex]\((a, b)\)[/tex] of the parallelogram given the three vertices [tex]\((-1, 2)\)[/tex], [tex]\((2, -1)\)[/tex], and [tex]\((3, 1)\)[/tex].
To start, let's define the given points:
- Point [tex]\(A\)[/tex] is [tex]\((-1, 2)\)[/tex]
- Point [tex]\(B\)[/tex] is [tex]\((2, -1)\)[/tex]
- Point [tex]\(C\)[/tex] is [tex]\((3, 1)\)[/tex]
In a parallelogram, the opposite sides are parallel and equal in length. This means we can use vector addition to find the fourth vertex. Specifically, the vector from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] added to the vector from [tex]\(A\)[/tex] to [tex]\(C\)[/tex] should give us the vector from [tex]\(A\)[/tex] to the fourth vertex [tex]\(D\)[/tex], which has coordinates [tex]\((a, b)\)[/tex].
First, calculate the vector [tex]\(\overrightarrow{AB}\)[/tex]:
[tex]\[
\overrightarrow{AB} = B - A = (2 - (-1), -1 - 2) = (3, -3)
\][/tex]
Next, calculate the vector [tex]\(\overrightarrow{AC}\)[/tex]:
[tex]\[
\overrightarrow{AC} = C - A = (3 - (-1), 1 - 2) = (4, -1)
\][/tex]
To find the coordinates of point [tex]\(D\)[/tex], we add the vectors [tex]\(\overrightarrow{AB}\)[/tex] and [tex]\(\overrightarrow{AC}\)[/tex] to point [tex]\(A\)[/tex]:
[tex]\[
D = A + \overrightarrow{AB} + \overrightarrow{AC}
\][/tex]
[tex]\[
D = (-1, 2) + (3, -3) + (4, -1)
\][/tex]
We perform the addition component-wise:
[tex]\[
x\text{-coordinate}: -1 + 3 + 4 = 6
\][/tex]
[tex]\[
y\text{-coordinate}: 2 - 3 - 1 = -2
\][/tex]
So, the coordinates of point [tex]\(D\)[/tex] are [tex]\((6, -2)\)[/tex]. Therefore, the coordinates for the fourth vertex [tex]\((a, b)\)[/tex] of the parallelogram are:
[tex]\[
(a, b) = (6, -2)
\][/tex]
