Question

plot the image of figure abcdefgh after a dilation with scale factor 2. one of the sides has been plotted for you. click twice to plot a segment. click a segment to delete it.

Explanation:

Step1: Identify Coordinates of Original Points

First, determine the coordinates of each vertex of the original figure \(ABCDEFGH\) using the grid. Let's assume the grid has a spacing of 1 unit. For example, if we take the bottom - left as a reference, we can find the coordinates of each point:

• Let's assume point \(G\) has coordinates \((x_G,y_G)\) and point \(H\) has coordinates \((x_H,y_H)\). From the plotted \(G'H'\), we know that dilation with scale factor \(k = 2\) means the coordinates of the image of a point \((x,y)\) after dilation centered at the origin (assuming the dilation is centered at the origin, if not, we can find the center of dilation. But since one side is plotted, we can use the relationship between \(G\), \(H\) and \(G'\), \(H'\)). If \(G'\) and \(H'\) are the images of \(G\) and \(H\) with scale factor 2, then if \(G=(x_G,y_G)\), \(G'=(2x_G,2y_G)\) and \(H=(x_H,y_H)\), \(H'=(2x_H,2y_H)\).
• For the other points: Let's find the coordinates of \(A\), \(B\), \(C\), \(D\), \(E\), \(F\). Suppose \(A=(x_A,y_A)\), \(B=(x_B,y_B)\), \(C=(x_C,y_C)\), \(D=(x_D,y_D)\), \(E=(x_E,y_E)\), \(F=(x_F,y_F)\).

Step2: Apply Dilation Formula

The formula for dilation with scale factor \(k\) centered at the origin is \((x,y)\to(kx,ky)\). If the center of dilation is not the origin, we first find the vector from the center of dilation to each point, scale the vector by \(k = 2\), and then add it back to the center of dilation. But since one side \(G'H'\) is already plotted, we can use the same center of dilation for all points.

• Let's assume the center of dilation is the same for all points. For each point \(P=(x,y)\) in \(ABCDEFGH\), the image \(P'=(2x,2y)\) (if centered at origin) or using the appropriate center - based formula.
• For example, if we consider the horizontal and vertical distances: If a side has length \(l\) in the original figure, the length of the side in the image after dilation with scale factor 2 will be \(2l\).
• Let's take point \(A\): If the coordinates of \(A\) are, say, \((x_A,y_A)\), then the coordinates of \(A'\) (the image of \(A\)) will be \((2x_A,2y_A)\) (assuming center of dilation is origin). Similarly, for point \(B\), \(B'=(2x_B,2y_B)\), and so on for all other points.

Step3: Plot the Image Points

Once we have the coordinates of all the image points \(A'\), \(B'\), \(C'\), \(D'\), \(E'\), \(F'\), \(G'\), \(H'\) (with \(G'\) and \(H'\) already partially plotted), we plot each point on the grid. For example, if \(A\) is at \((a,b)\), \(A'\) will be at \((2a,2b)\). We then connect the points in the same order as the original figure \(ABCDEFGH\) to get the dilated figure \(A'B'C'D'E'F'G'H'\).

(Note: Since this is a plotting problem, the key is to use the dilation formula \( (x,y)\to(2x,2y) \) (or the appropriate center - based dilation) to find the coordinates of each vertex's image and then plot them. The following is a general description of the process. In a practical grid - based plotting, we would:

1. Find the coordinates of each vertex of \(ABCDEFGH\) by looking at their positions on the grid (e.g., count the number of units from the x - axis and y - axis).
2. Multiply each coordinate by 2 to get the coordinates of the image vertices.
3. Plot each image vertex and connect them in the correct order.)

Answer:

To plot the dilated figure \(A'B'C'D'E'F'G'H'\):

1. Determine the coordinates of each vertex of \(ABCDEFGH\) from the grid.
2. For each vertex \((x,y)\), calculate the image vertex as \((2x,2y)\) (assuming dilation centered at the origin; adjust if the center of dilation is different, but since \(G'H'\) is plotted, use the same center logic).
3. Plot the image vertices \(A'\), \(B'\), \(C'\), \(D'\), \(E'\), \(F'\), \(G'\), \(H'\) (with \(G'\) and \(H'\) already having a part plotted) and connect them in the order of the original figure.

(In a digital plotting tool, you would click on the calculated coordinates of each image vertex to plot the segments between them, following the shape of \(ABCDEFGH\) but with each side length doubled.)