Question

the coordinates of three vertices of a rectangle are (3,7), (-3,5), and (0,-4). what are the coordinates of the fourth vertex? a. (6,-2) b. (6,2) c. (-2,-6) d. (-2,6)

Explanation:

Step1: Recall the property of rectangle

In a rectangle, the diagonals bisect each other. Let the vertices of the rectangle be \(A(3,7)\), \(B(- 3,5)\), \(C(0,-4)\) and \(D(x,y)\). The mid - point of diagonal \(AC\) is the same as the mid - point of diagonal \(BD\).
The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
The mid - point of \(AC\) with \(A(3,7)\) and \(C(0,-4)\) is \((\frac{3 + 0}{2},\frac{7+( - 4)}{2})=(\frac{3}{2},\frac{3}{2})\).
The mid - point of \(BD\) with \(B(-3,5)\) and \(D(x,y)\) is \((\frac{-3 + x}{2},\frac{5 + y}{2})\).

Step2: Set up equations for mid - points

We have the following two equations based on the equality of mid - points:
\(\frac{-3 + x}{2}=\frac{3}{2}\) and \(\frac{5 + y}{2}=\frac{3}{2}\).
For \(\frac{-3 + x}{2}=\frac{3}{2}\), cross - multiply: \(-3+x = 3\), then \(x=6\).
For \(\frac{5 + y}{2}=\frac{3}{2}\), cross - multiply: \(5 + y=3\), then \(y=-2\).

Answer:

A. \((6,-2)\)