Question

1. given the points a(-2,1), b(-6,-9), c(-1,-11) on the coordinates axes below. state the coordinates of point d such that a, b, c, and d would form a rectangle.

Explanation:

Step1: Recall property of rectangle

In a rectangle, the diagonals bisect each other. Let the coordinates of point D be \((x,y)\). The mid - point of diagonal AC and the mid - point of diagonal BD are the same.
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 where \(A(-2,1)\) and \(C(-1,-11)\) is \((\frac{-2+( - 1)}{2},\frac{1+( - 11)}{2})=(\frac{-2 - 1}{2},\frac{1 - 11}{2})=(-\frac{3}{2},-5)\)
The mid - point of BD where \(B(-6,-9)\) and \(D(x,y)\) is \((\frac{-6 + x}{2},\frac{-9 + y}{2})\)

Step2: Set up equations

Since the mid - points of the diagonals are equal, we have the following two equations:
\(\frac{-6 + x}{2}=-\frac{3}{2}\) and \(\frac{-9 + y}{2}=-5\)
For \(\frac{-6 + x}{2}=-\frac{3}{2}\), multiply both sides by 2: \(-6+x=-3\), then \(x=-3 + 6=3\)
For \(\frac{-9 + y}{2}=-5\), multiply both sides by 2: \(-9 + y=-10\), then \(y=-10 + 9=-1\)

Answer:

\((3,-1)\)