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The image shows three vertices of a parallelogram, and the problem asks to find the coordinates of the fourth vertex. The known vertices are:

- \( (-6, -4) \)
- \( (-1, 1) \)
- \( (2, 0) \)

To find the fourth vertex, we can use the property of a parallelogram that the diagonals bisect each other. This means the midpoint of one diagonal will be the same as the midpoint of the other diagonal.

Let's solve the problem step-by-step:

1. Calculate the midpoint of the diagonal between \( (-6, -4) \) and the unknown point \( (x, y) \), the fourth vertex.
2. Calculate the midpoint of the diagonal between \( (-1, 1) \) and \( (2, 0) \).
3. Equate the midpoints and solve for \( (x, y) \), the coordinates of the fourth vertex.

Let me proceed with these calculations.The coordinates of the fourth vertex of the parallelogram are \( (7, 5) \).

Would you like a detailed explanation or have any questions about the solution?

Here are 5 related questions to expand your understanding:
1. How can you find the midpoint between any two points in the coordinate plane?
2. What are other properties of parallelograms that can help solve geometric problems?
3. How can we verify that four points form a parallelogram using vector geometry?
4. How would the approach differ if we had a rectangle instead of a general parallelogram?
5. Can the method of diagonals be used to find the fourth vertex of any quadrilateral?

**Tip:** In a parallelogram, opposite sides are equal and parallel, which can be useful when solving problems involving unknown vertices.

En la siguiente figura se muestran tres vértices de un paralelogramo. Hallar las coordenadas del cuarto vértice.

Solve a problem involving a parallelogram with three known vertices (-6, -4), (-1, 1), and (2, 0). Learn how to find the fourth vertex using the midpoint formula and the properties of parallelograms. Ideal for students in Grades 9-11.

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