Question

three vertices of a parallelogram are shown in the figure below. give the coordinates of the fourth vertex.

Explanation:

Step1: Recall property of parallelogram

In a parallelogram, the mid - points of the diagonals coincide. Let the vertices be \(A(-6,-5)\), \(B(0, - 3)\), \(C(3,6)\) and the fourth vertex be \(D(x,y)\). The diagonals of a parallelogram bisect each other. If we consider the diagonals \(AC\) and \(BD\), the mid - point of \(AC\) is the same as the mid - point of \(BD\).
The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
The mid - point of \(AC\) with \(A(-6,-5)\) and \(C(3,6)\) is \((\frac{-6 + 3}{2},\frac{-5+6}{2})=(-\frac{3}{2},\frac{1}{2})\).
The mid - point of \(BD\) with \(B(0,-3)\) and \(D(x,y)\) is \((\frac{0 + x}{2},\frac{-3 + y}{2})\).

Step2: Equate the x - coordinates of mid - points

\(\frac{0 + x}{2}=-\frac{3}{2}\), then \(x=-3\).

Step3: Equate the y - coordinates of mid - points

\(\frac{-3 + y}{2}=\frac{1}{2}\), then \(-3 + y = 1\), and \(y=4\).

Answer:

\((-3,4)\)