To find the fourth vertex of the rectangle, we have the three given vertices at \( (5, 2) \), \( (8, 2) \), and \( (5, -5) \).
Let's analyze the vertices:
-
The points \( (5, 2) \) and \( (8, 2) \) share the same \( y \)-coordinate, which means they are on a horizontal line. The distance between them is: \[ 8 - 5 = 3 \] Thus, they represent one side of the rectangle, which is 3 units long.
-
The point \( (5, -5) \) shares the same \( x \)-coordinate as \( (5, 2) \). This means that \( (5, -5) \) is vertically aligned with \( (5, 2) \).
Since \( (5, 2) \) and \( (8, 2) \) are the horizontal corners of the rectangle, we can determine the fourth vertex by taking the horizontal distance from \( (5, 2) \) to be the same vertical distance from \( (5, -5) \).
The vertical distance from \( (5, 2) \) to \( (5, -5) \) is: \[ 2 - (-5) = 7 \]
The fourth vertex must be horizontally aligned with \( (8, 2) \) and vertically aligned with \( (5, -5) \).
Thus, the fourth vertex will have the same \( y \)-coordinate as \( (5, -5) \) and the same \( x \)-coordinate as \( (8, 2) \): \[ (x, y) = (8, -5) \]
Therefore, the coordinates of the fourth vertex are \( (8, -5) \).
Now, let's find the area of the rectangle. The lengths of the sides are:
- Width (between \( (5, 2) \) and \( (8, 2) \)): 3 units.
- Height (between \( (5, 2) \) and \( (5, -5) \)): 7 units.
The area \( A \) of the rectangle is given by: \[ A = \text{width} \times \text{height} \] \[ A = 3 \times 7 = 21 \]
Thus, the coordinates of the fourth vertex are \( (8, -5) \), and the area of the rectangle is \( 21 \) square units.