Question

1. what location would a point need to be plotted to form a rectangle with the three points already on the coordinate plane? (a) (-4, 3) (b) (-3, 2) (c) (4, - 2) (d) (2, -3)

Explanation:

Step1: Recall rectangle property

In a rectangle, opposite sides are equal and parallel. The given points form two - side lengths of the rectangle. We can use the mid - point formula concept (diagonals of a rectangle bisect each other). If the three given points are \(A(x_1,y_1)\), \(B(x_2,y_2)\) and \(C(x_3,y_3)\) and the fourth point is \(D(x_4,y_4)\), the mid - point of \(AC\) is the same as the mid - point of \(BD\). Another way is to observe the horizontal and vertical distances between the given points.
Let the three given points be \((- 3,-2)\), \((3,-2)\) and \((3,2)\).
The horizontal distance between \((-3,-2)\) and \((3,-2)\) is \(d_x=\vert3-( - 3)\vert = 6\).
The vertical distance between \((3,-2)\) and \((3,2)\) is \(d_y=\vert2-( - 2)\vert=4\).
The fourth point should have the same horizontal and vertical relationships with the other points.
If we consider the point \((-3,-2)\) and \((3,2)\), the fourth point's \(x\) - coordinate is obtained by using the \(x\) - coordinates of the non - collinear points and the \(y\) - coordinate is obtained in a similar way.
The fourth point's \(x\) - coordinate: If we consider the \(x\) - values of the non - collinear points \((-3,-2)\) and \((3,2)\), the \(x\) - value of the fourth point is \(-3+(3 - 3)=-3\).
The fourth point's \(y\) - coordinate: If we consider the \(y\) - values of the non - collinear points \((-3,-2)\) and \((3,2)\), the \(y\) - value of the fourth point is \(2+( - 2-(-2)) = 2\). So the point is \((-3,2)\).

Step2: Check the options

We check each option against our calculated point. Option (b) is \((-3,2)\).

Answer:

B. \((-3,2)\)