Question

A rectangle in the coordinate plane has vertices at (5,8), (2,4), and (5,4). What are the coordinates of the fourth vertex?

Answer 1

To find the coordinates of the fourth vertex of the rectangle, we first note the given vertices: \( A = (5, 8) \), \( B = (2, 4) \), and \( C = (5, 4) \).

1. **Identify the known vertices**:
- Point \( A = (5, 8) \) is located at \( x = 5 \) and \( y = 8 \).
- Point \( B = (2, 4) \) is located at \( x = 2 \) and \( y = 4 \).
- Point \( C = (5, 4) \) is located at \( x = 5 \) and \( y = 4 \).

2. **Find the coordinates of the fourth vertex \( D \)**:
- The rectangle's opposite vertices must share the same \( x \)- or \( y \)-coordinates.
- Let’s compare the coordinates:
- Point \( A \) has the same \( x \)-coordinate as point \( C \) (both have \( x = 5 \)).
- Point \( B \) has a different \( x \)-coordinate.

Now, we see that:

- The \( y \)-coordinate for \( C \) is \( 4 \), and the \( y \)-coordinate for \( A \) is \( 8 \).
- Thus, the fourth vertex \( D \) must have the same \( y \)-coordinate as \( B \) (which is \( 4 \)), and the same \( x \)-coordinate as \( B \) (which is \( 2 \)).

Therefore, the coordinates of the fourth vertex \( D \) are:

\[
D = (2, 8)
\]

3. **Conclusion**: The coordinates of the fourth vertex of the rectangle are \( (2, 8) \).