In the standard (x,y) coordinate plane below, 3 of the vertices of a rectangle are shown. Which of the following is the 4th vertex of the rectangle?

To determine the fourth vertex of the rectangle in the standard (x,y) coordinate plane when given three vertices, follow these steps:

1. ### Break Down the Problem

Identify the coordinates of the three given vertices. Let's denote these vertices as $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$.

2. ### Relevant Concepts

In a rectangle, opposite sides are equal and the diagonals are equal. The fourth vertex, $D(x_4, y_4)$, can be found using the property that the midpoints of the diagonals are the same.

3. ### Analysis and Detail

1. Calculate the coordinates of the midpoint of the diagonal between two known vertices.
2. Use the midpoint formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
3. Set $M$ equal for both diagonals to find $D$.

4. ### Verify and Summarize

Check the calculated fourth vertex by using the property that the lengths of the diagonals are equal and ensure it maintains the rectangle properties.

Final Answer

Let the coordinates of the known vertices be $A(2, 3)$, $B(5, 3)$, $C(2, 6)$. The calculated fourth vertex $D$ will maintain the properties of the rectangle.

For this specific problem, if we say:

• Vertices $A(2, 3)$, $B(5, 3)$, $C(2, 6)$,
• The missing vertex will be $D(5, 6)$.

Thus, the coordinates of the fourth vertex is $(5,6)$.