To find the fourth vertex of a parallelogram when three vertices are given, we can use the properties of a parallelogram. Specifically, opposite vertices of a parallelogram add up to twice the coordinates of the midpoint. Given the vertices [tex]\(A = (2,1)\)[/tex], [tex]\(B = (4,7)\)[/tex], and [tex]\(C = (6,5)\)[/tex], we need to find [tex]\(D\)[/tex].
Let's denote the fourth vertex by [tex]\(D(x, y)\)[/tex]. There are three possible scenarios for finding the coordinates of [tex]\(D\)[/tex]:
### Case 1: [tex]\(A\)[/tex] and [tex]\(B\)[/tex] as one pair of opposite vertices, [tex]\(C\)[/tex] and [tex]\(D\)[/tex] as the other
We can find the fourth vertex [tex]\(D\)[/tex] using the following method:
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(C\)[/tex] by the same vector that translates [tex]\(A\)[/tex] to [tex]\(B\)[/tex].
The translation vector from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] is:
[tex]\[ (B_x - A_x, B_y - A_y) = (4 - 2, 7 - 1) = (2, 6) \][/tex]
Now we apply this translation to [tex]\(C\)[/tex]:
[tex]\[ D = (C_x + (B_x - A_x), C_y + (B_y - A_y)) = (6 + 2, 5 + 6) = (8, 11) \][/tex]
### Case 2: [tex]\(A\)[/tex] and [tex]\(C\)[/tex] as one pair of opposite vertices, [tex]\(B\)[/tex] and [tex]\(D\)[/tex] as the other
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(B\)[/tex] by the same vector that translates [tex]\(A\)[/tex] to [tex]\(C\)[/tex].
The translation vector from [tex]\(A\)[/tex] to [tex]\(C\)[/tex] is:
[tex]\[ (C_x - A_x, C_y - A_y) = (6 - 2, 5 - 1) = (4, 4) \][/tex]
Now we apply this translation to [tex]\(B\)[/tex]:
[tex]\[ D = (B_x + (C_x - A_x), B_y + (C_y - A_y)) = (4 + 4, 7 + 4) = (8, 11) \][/tex]
### Case 3: [tex]\(B\)[/tex] and [tex]\(C\)[/tex] as one pair of opposite vertices, [tex]\(A\)[/tex] and [tex]\(D\)[/tex] as the other
- The coordinate of [tex]\(D\)[/tex] can be obtained by translating [tex]\(A\)[/tex] by the same vector that translates [tex]\(B\)[/tex] to [tex]\(C\)[/tex].
The translation vector from [tex]\(B\)[/tex] to [tex]\(C\)[/tex] is:
[tex]\[ (C_x - B_x, C_y - B_y) = (6 - 4, 5 - 7) = (2, -2) \][/tex]
Now we apply this translation to [tex]\(A\)[/tex]:
[tex]\[ D = (A_x + (C_x - B_x), A_y + (C_y - B_y)) = (2 + 2, 1 + (-2)) = (4, -1) \][/tex]
### Summary:
The three possible coordinates for the fourth vertex [tex]\(D\)[/tex] are:
- [tex]\( (8, 11) \)[/tex]
- [tex]\( (8, 11) \)[/tex]
- [tex]\( (4, -1) \)[/tex]
Thus, the three possible coordinates from the given options are:
[tex]\[
\boxed{(8, 11), (8, 11), (4, -1)}
\][/tex]
