The coordinates of three vertices of a rectangle are \( (3,7),(-3,5) \), and \( (0,-4) \). What are the coordinates of the fourth vertex? O A. \( \begin{array}{ll}(6,2) \\ \text { O. } & (6,-2) \\ \text { O. } & (-2,-6) \\ \text { D. } & (-2,6) \\ \text { Reset } & \text { Next }\end{array} \)

To find the coordinates of the fourth vertex of the rectangle, we first need to identify the coordinates of the given vertices. Let’s denote the vertices as \( A(3,7) \), \( B(-3,5) \), and \( C(0,-4) \). A rectangle has sides that are either parallel to the coordinate axes or form angles of 90 degrees with each other. We can use the property that the diagonals of a rectangle bisect each other. The midpoint of diagonal \( AC \) should equal the midpoint of diagonal \( BD \), where \( D \) is the unknown fourth vertex. Calculating the midpoint of \( A \) and \( C \): \[ \text{Midpoint of } AC = \left( \frac{3 + 0}{2}, \frac{7 + (-4)}{2} \right) = \left( \frac{3}{2}, \frac{3}{2} \right) \] With the coordinates of point \( B(-3,5) \) known, we can set up the equations for the midpoint of \( BD \) with \( D(x,y) \): \[ \text{Midpoint of } BD = \left( \frac{-3 + x}{2}, \frac{5 + y}{2} \right) \] Setting the midpoints equal gives us the equations: \[ \frac{-3 + x}{2} = \frac{3}{2} \quad \text{and} \quad \frac{5 + y}{2} = \frac{3}{2} \] Solving these equations: 1. \( -3 + x = 3 \) → \( x = 6 \) 2. \( 5 + y = 3 \) → \( y = -2 \) Therefore, the coordinates of the fourth vertex \( D \) are \( (6, -2) \). So, the answer is \( (6, -2) \).