Question

you are graphing rectangle abcd in the coordinate plane. the following are three of the vertices of the rectangle: a(3, 5), b(3, -4), and c(-2, -4). what are the coordinates of point d? (□,□)

Explanation:

Step1: Analyze the properties of a rectangle

In a rectangle, opposite sides are equal and parallel. Also, \(AB\) is a vertical line (since \(x -\)coordinate of \(A(3,5)\) and \(B(3, - 4)\) is the same, \(x = 3\)). \(BC\) is a horizontal line (since \(y -\)coordinate of \(B(3,-4)\) and \(C(-2,-4)\) is the same, \(y=-4\)).

Step2: Determine the \(x -\)coordinate of \(D\)

Since \(AD\) is parallel to \(BC\), and \(BC\) has a change in \(x -\)coordinate from \(x = 3\) (for \(B\)) to \(x=-2\) (for \(C\)). The \(x -\)coordinate of \(A\) is \(x = 3\). Let the \(x -\)coordinate of \(D\) be \(x\). Using the property that \(AD\parallel BC\), we know that the change in \(x\) from \(A\) to \(D\) is the same as the change in \(x\) from \(B\) to \(C\). The change in \(x\) for \(BC\) is \(\Delta x=-2 - 3=-5\). So \(x=3-5=-2\)

Step3: Determine the \(y -\)coordinate of \(D\)

Since \(CD\) is parallel to \(AB\). The \(y -\)coordinate of \(C\) is \(y=-4\). The change in \(y\) for \(AB\) is \(5-(-4)=9\). Using the property that \(CD\parallel AB\), the \(y -\)coordinate of \(D\) is \(y=-4 + 9=5\)

Answer:

\((-2,5)\)