The problem asks for the coordinates of the fourth vertex of a parallelogram given the coordinates of three vertices:

1. (-2, -7)
2. (6, -2)
3. (9, -9)

Solving the Problem

The defining property of a parallelogram is that its opposite sides are parallel and equal in length.

1. Let’s denote the fourth vertex as $(x, y)$.
2. To find $(x, y)$, consider that the diagonal divides the parallelogram into two congruent triangles. The midpoints of the diagonals of a parallelogram coincide.

Steps:

• The midpoint formula is: $\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
• The diagonal between points $(-2, -7)$ and $(9, -9)$ should have the same midpoint as the diagonal between points $(6, -2)$ and $(x, y)$.

Calculate the midpoint of diagonal 1:

$\text{Midpoint} = \left( \frac{-2 + 9}{2}, \frac{-7 + (-9)}{2} \right) = \left( \frac{7}{2}, -8 \right)$

Set the midpoint of diagonal 2 equal to the above midpoint:

$\left( \frac{6 + x}{2}, \frac{-2 + y}{2} \right) = \left( \frac{7}{2}, -8 \right)$

From here, solve for $x$ and $y$:

1. For the x-coordinate: $\frac{6 + x}{2} = \frac{7}{2} \implies 6 + x = 7 \implies x = 1$
2. For the y-coordinate: $\frac{-2 + y}{2} = -8 \implies -2 + y = -16 \implies y = -14$

Final Answer:

The fourth vertex is $(1, -14)$.

Would you like additional details or clarifications?

5 Related Questions:

1. How do you verify the properties of a parallelogram using coordinates?
2. What are other methods to find the fourth vertex of a parallelogram?
3. How does the midpoint formula relate to the diagonals of polygons?
4. Can this method be generalized to other quadrilaterals?
5. How can we calculate the area of this parallelogram?

Tip:

Always verify your result by checking if opposite sides are equal and parallel!