three point of a parallelogram are 5,0 9,0 3,3 what's the forth point

To find the fourth point of a parallelogram when three points are given, we can use the property that opposite sides of a parallelogram are equal and parallel. Let's denote the given points as follows:

- \( A(5, 0) \)
- \( B(9, 0) \)
- \( C(3, 3) \)

We need to find the fourth point \( D \). A common approach is to consider two possible scenarios for where the point \( D \) might be:

### 1. \( D \) Opposite to \( A \) with \( B \) and \( C \) being adjacent vertices:
In this case, the pairs of opposite sides are \( AB \) and \( CD \), and \( AC \) and \( BD \).

- Vector \( AB = B - A = (9 - 5, 0 - 0) = (4, 0) \)
- Vector \( AC = C - A = (3 - 5, 3 - 0) = (-2, 3) \)

Since \( AB \) and \( CD \) are opposite sides and equal, the vector from \( C \) to \( D \) should be equal to vector \( AB \):
- So, if \( D = (x, y) \), then \( D = C + AB \)
- \( D = (3, 3) + (4, 0) = (7, 3) \)

### 2. \( D \) Opposite to \( B \) with \( A \) and \( C \) being adjacent vertices:
In this case, the pairs of opposite sides are \( AD \) and \( BC \), and \( AB \) and \( CD \).

- Vector \( BC = C - B = (3 - 9, 3 - 0) = (-6, 3) \)

Since \( AD \) and \( BC \) are opposite sides and equal, the vector from \( A \) to \( D \) should be equal to vector \( BC \):
- So, if \( D = (x, y) \), then \( D = A + BC \)
- \( D = (5, 0) + (-6, 3) = (-1, 3) \)

### Conclusion
The two possible fourth points of the parallelogram can be \( D(7, 3) \) or \( D(-1, 3) \).

To determine which one forms a parallelogram, let's check:

- \( A(5, 0) \), \( B(9, 0) \), \( C(3, 3) \), \( D(7, 3) \) form a parallelogram, since \( AB = CD = (4, 0) \) and \( AC = BD = (-2, 3) \).
- The point \( D(-1, 3) \) forms a different set of opposite sides and does not fit the original problem's requirement.

Therefore, the fourth point \( D \) that completes the parallelogram is \( D(7, 3) \).

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### Related Questions:
1. How do you find the centroid of a parallelogram given its vertices?
2. How can you verify if four given points form a parallelogram?
3. What is the formula to find the area of a parallelogram using its vertices?
4. How do you find the coordinates of the midpoints of the sides of a parallelogram?
5. What are the properties of a parallelogram concerning its diagonals?

**Tip:** When dealing with coordinate geometry problems, sketching the points on a graph can help visualize the solution.

Learn how to find the fourth point of a parallelogram when three vertices are given as (5,0), (9,0), and (3,3). This problem involves applying coordinate geometry principles and parallelogram properties to determine the correct fourth vertex. Suitable for high school students studying coordinate geometry and vector mathematics.

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