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 $D(7, 3)$.

Would you like more details or have any questions?

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.