Question

for the figure below, do a dilation centered at the origin with a scale factor of $\frac{1}{2}$. then answer the questions. (a) find area of original figure: square units area of final figure: square units (b) fill in the blank to make a true statement. area of final figure $=\square \times$ area of original figure (c) true or false? the original figure and the final figure are not similar. true false

Explanation:

Step1: Find the area of the original figure

The original figure is a rectangle. The formula for the area of a rectangle is \(A = l\times w\). From the graph, the length \(l = 6\) units (from \(y = 2\) to \(y = 8\)) and the width \(w=2\) units (from \(x = 2\) to \(x = 4\)). So, \(A_{original}=6\times2 = 12\) square units.

Step2: Find the dimensions of the dilated figure

For a dilation centered at the origin with a scale factor \(k=\frac{1}{2}\), if the original length \(l\) and width \(w\), the new length \(l'=k\times l\) and new width \(w'=k\times w\). So \(l'=\frac{1}{2}\times6 = 3\) and \(w'=\frac{1}{2}\times2 = 1\).

Step3: Find the area of the dilated figure

Using the area formula \(A = l\times w\) for the dilated rectangle, \(A_{final}=3\times1=3\) square units.

Step4: Find the relationship between the areas

We know that \(A_{final}=3\) and \(A_{original} = 12\). Let \(A_{final}=m\times A_{original}\). Then \(m=\frac{A_{final}}{A_{original}}=\frac{3}{12}=\frac{1}{4}\).

Step5: Determine similarity

Two figures are similar if their corresponding angles are equal (for rectangles, all angles are \(90^{\circ}\)) and the ratios of their corresponding sides are equal. For dilation, the ratio of corresponding sides is the scale factor. Since dilation preserves the shape (angles remain the same) and the ratio of sides is constant (\(\frac{1}{2}\)), the original and dilated figures are similar.

Answer:

(a) Area of original figure: \(12\) square units; Area of final figure: \(3\) square units.
(b) Area of final figure \(=\frac{1}{4}\times\) Area of original figure.
(c) False.