Given that we have a parallelogram with two known vertices at [tex]\((8, 6)\)[/tex] and [tex]\((6, 0)\)[/tex], and the point of intersection of the diagonals at [tex]\((2, 4)\)[/tex]. Let's find the coordinates of the remaining vertices.
1. Understanding the properties of the parallelogram:
- The diagonals of a parallelogram bisect each other.
- The point of intersection of the diagonals is the midpoint of each diagonal.
2. Identify known points:
- [tex]\( A = (8, 6) \)[/tex]
- [tex]\( B = (6, 0) \)[/tex]
- Intersection point (midpoint of the diagonals), [tex]\( M = (2, 4) \)[/tex]
3. Let [tex]\( C \)[/tex] and [tex]\( D \)[/tex] be the unknown vertices of the parallelogram:
- We need to determine the coordinates of [tex]\( C \)[/tex] and [tex]\( D \)[/tex].
4. Using the midpoint formula for diagonals:
- [tex]\( M \)[/tex] is the midpoint of both [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex].
- When two points [tex]\( A \)[/tex] and [tex]\( C \)[/tex] have their midpoint [tex]\( M \)[/tex]:
[tex]\[
M = \left( \frac{A_x + C_x}{2}, \frac{A_y + C_y}{2} \right)
\][/tex]
- Using the intersection point [tex]\( M = (2, 4) \)[/tex]:
[tex]\[
2 = \frac{8 + C_x}{2} \quad \text{and} \quad 4 = \frac{6 + C_y}{2}
\][/tex]
5. Solving for [tex]\( C_x \)[/tex] and [tex]\( C_y \)[/tex]:
[tex]\[
2 = \frac{8 + C_x}{2} \implies 4 = 8 + C_x \implies C_x = 4 - 8 \implies C_x = -4
\][/tex]
[tex]\[
4 = \frac{6 + C_y}{2} \implies 8 = 6 + C_y \implies C_y = 8 - 6 \implies C_y = 2
\][/tex]
- Thus, [tex]\( C = (-4, 2) \)[/tex]
6. Using midpoint formula for [tex]\(B\)[/tex] and [tex]\(D\)[/tex]:
[tex]\[
2 = \frac{6 + D_x}{2} \quad \text{and} \quad 4 = \frac{0 + D_y}{2}
\][/tex]
7. Solving for [tex]\( D_x \)[/tex] and [tex]\( D_y \)[/tex]:
[tex]\[
2 = \frac{6 + D_x}{2} \implies 4 = 6 + D_x \implies D_x = 4 - 6 \implies D_x = -2
\][/tex]
[tex]\[
4 = \frac{0 + D_y}{2} \implies 8 = 0 + D_y \implies D_y = 8
\][/tex]
- Thus, [tex]\( D = (-2, 8) \)[/tex]
Therefore, the coordinates of the remaining vertices of the parallelogram are [tex]\( (-4, 2) \)[/tex] and [tex]\( (-2, 8) \)[/tex].
