Question

plot the image of figure abcd after a dilation with scale factor 3. one of the sides has been plotted for you.
click twice to plot a segment.
click a segment to delete it.

Explanation:

Step1: Determine the center of dilation

Assume the center of dilation is the origin (since no other information is given). For a dilation with scale factor \(k = 3\), if a point \(P(x,y)\) is dilated, the image \(P'(x',y')\) is given by \(x'=kx\) and \(y' = ky\).

Step2: Find the coordinates of \(C'\) and \(D'\)

Let's assume \(A=(x_1,y_1)\), \(B=(x_2,y_2)\), \(C=(x_3,y_3)\), \(D=(x_4,y_4)\). Since \(A'B'\) is already plotted (after dilation of \(AB\)), for side \(BC\) (vertical side in the original square \(ABCD\)), if \(B=(x_2,y_2)\) and \(C=(x_2,y_2 - a)\) (assuming length of \(BC=a\)), then \(B'=(3x_2,3y_2)\) and \(C'=(3x_2,3(y_2 - a))\). Similarly, for side \(AD\) (vertical side in the original square \(ABCD\)), if \(A=(x_1,y_1)\) and \(D=(x_1,y_1 - a)\), then \(A'=(3x_1,3y_1)\) and \(D'=(3x_1,3(y_1 - a))\). Then connect \(B'\) to \(C'\), \(C'\) to \(D'\), and \(D'\) to \(A'\) to form the dilated square \(A'B'C'D'\).

Since we can't actually click on a digital - like interface here, but conceptually:

• If \(AB\) is a horizontal side of length \(l\) (in terms of grid - units) in the original figure \(ABCD\), \(A'B'\) has length \(3l\) (already done).
• The vertical sides \(BC\) and \(AD\) of length \(m\) (in terms of grid - units) in \(ABCD\) will have images \(B'C'\) and \(A'D'\) of length \(3m\). We plot \(C'\) such that \(B'C'\) is parallel to \(BC\) (vertical if \(BC\) is vertical) and three times as long, and \(D'\) such that \(A'D'\) is parallel to \(AD\) (vertical if \(AD\) is vertical) and three times as long, then connect the points.

Answer:

Plot \(C'\) and \(D'\) such that \(B'C'\) and \(A'D'\) are vertical (assuming original \(BC\) and \(AD\) are vertical) with length three times the length of \(BC\) and \(AD\) respectively, then connect \(B'\) to \(C'\), \(C'\) to \(D'\), and \(D'\) to \(A'\) to form the dilated square \(A'B'C'D'\).