Question

complete problems 3 and 4. in woodworking class, eduardo makes a pattern for the top of a wooden box. he uses scale copies of figure j. the center of dilation is point m. a. eduardo uses a scale factor of 2 to draw figure k. complete figure k by drawing the missing sides. b. eduardo uses a scale factor of $\frac{1}{2}$ to draw figure l. complete figure l by drawing the missing sides. a wallpaper designer draws figure c as the original figure in a wallpaper pattern. then they draw figures d and e. the center of dilation is point f. which figures, if any, are dilations of figure c? show your work.

Explanation:

Step1: Recall dilation rules

For a dilation with scale - factor \(k\), the distance from the center of dilation to each point in the new figure is \(k\) times the distance from the center of dilation to the corresponding point in the original figure.

Step2: Solve part a

When the scale - factor \(k = 2\), for each side in the original figure, the corresponding side in figure \(K\) will be twice as long. Measure the distances of the sides of the original figure from the center of dilation \(M\) and then draw the sides of figure \(K\) such that their distances from \(M\) are twice those of the original - figure sides.

Step3: Solve part b

When the scale - factor \(k=\frac{1}{2}\), for each side in the original figure, the corresponding side in figure \(L\) will be half as long. Measure the distances of the sides of the original figure from the center of dilation \(M\) and then draw the sides of figure \(L\) such that their distances from \(M\) are half those of the original - figure sides.

Step4: Analyze dilations for the wallpaper figures

To determine if figures \(D\) and \(E\) are dilations of figure \(C\) with center of dilation \(F\), check if the ratios of the distances of corresponding points from \(F\) are constant. Measure the distances from \(F\) to corresponding points on figures \(C\), \(D\), and \(E\). If the ratio of the distance from \(F\) to a point on figure \(D\) (or \(E\)) and the distance from \(F\) to the corresponding point on figure \(C\) is the same for all corresponding points, then the figure is a dilation of figure \(C\).

Since this is a drawing - based problem, the actual drawing of the figures \(K\) and \(L\) and the determination of dilations for figures \(C\), \(D\), and \(E\) need to be done on the provided grid or figure. Without actually drawing, we can't provide the final visual answers. But the steps above outline the process for solving the problem.

Answer:

For part a: Draw figure \(K\) with sides twice as long as the corresponding sides of the original figure with respect to the center of dilation \(M\).
For part b: Draw figure \(L\) with sides half as long as the corresponding sides of the original figure with respect to the center of dilation \(M\).
For the wallpaper figures: Measure the distances from \(F\) to corresponding points on \(C\), \(D\), and \(E\). If the ratios of the distances are constant for all corresponding points, then the figure is a dilation of \(C\).