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: Recall Dilation Rules

Dilation with scale factor \( k \) multiplies each coordinate of a point \((x,y)\) by \( k \). So for scale factor 2, new coordinates \((x',y') = (2x, 2y)\).

Step2: Identify Original Coordinates (Assume Grid Units)

• Let's assign coordinates (assuming each grid dot is 1 unit). For example:
• Point \( C \): Let's say \( C = (x_C, y_C) \), \( B = (x_B, y_B) \), \( D = (x_D, y_D) \), \( E = (x_E, y_E) \), \( F = (x_F, y_F) \), \( G = (x_G, y_G) \), \( H = (x_H, y_H) \), \( A = (x_A, y_A) \).
• Given \( F'G' \) is plotted, so original \( F \) and \( G \) were dilated to \( F' \) and \( G' \). Let's find original \( F \): If \( F' \) is at some \( (x_{F'}, y_{F'}) \), then original \( F = (\frac{x_{F'}}{2}, \frac{y_{F'}}{2}) \), same for \( G \).

Step3: Dilate Each Point

• For each vertex of \( ABCDEFGH \), multiply its \( x \) and \( y \) coordinates by 2.
• Example: If \( C = (x_1, y_1) \), then \( C' = (2x_1, 2y_1) \); \( B = (x_2, y_2) \), \( B' = (2x_2, 2y_2) \), etc.
• Plot the new points \( A', B', C', D', E', F', G', H' \) and connect them as per the original shape's edges.

(Note: Since the grid is visual, the key is to apply the scale factor 2 to each vertex's coordinates. For example, if a side from \( D \) to \( E \) is length \( l \), after dilation it's \( 2l \), and the direction remains the same as dilation is a similarity transformation.)

Answer:

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

1. Identify Coordinates: For each vertex of \( ABCDEFGH \), determine its \((x, y)\) coordinates (using the grid).
2. Apply Dilation (Scale Factor 2): Multiply each \( x \)- and \( y \)-coordinate by 2 to get the new coordinates \((2x, 2y)\) for each vertex.
3. Plot New Points: Mark the new vertices \( A', B', C', D', E', F', G', H' \) on the grid using the dilated coordinates.
4. Connect Edges: Draw segments between the new vertices in the same order as the original figure (e.g., \( A' \) to \( B' \), \( B' \) to \( C' \), etc.), matching the original shape’s edge connections.

(Visually, each side length doubles, and the figure remains similar to \( ABCDEFGH \), centered around the same center of dilation (implied as the origin or the center relative to the grid, but typically dilation from the origin or a fixed center; here, since \( F'G' \) is plotted, the center is consistent for all points).)