Question

for the figure below, do a dilation centered at the origin with a scale factor of 3. then answer the questions. (a) finc 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 = × 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. From the graph, its length \(l = 4 - 2=2\) units and its width \(w = 3 - 2 = 1\) unit.
Using the formula for the area of a rectangle \(A = l\times w\), we have \(A_{original}=2\times1 = 2\) square units.

Step2: Find the dimensions of the dilated figure

For a dilation centered at the origin with a scale factor \(k = 3\), if the original length is \(l\) and width is \(w\), the new length \(l_{new}=k\times l\) and new width \(w_{new}=k\times w\).
So \(l_{new}=3\times2 = 6\) units and \(w_{new}=3\times1=3\) units.

Step3: Find the area of the dilated figure

Using the area formula \(A = l\times w\) for the dilated rectangle, \(A_{final}=6\times3=18\) square units.

Step4: Find the relationship between the areas

We know that \(A_{final}=18\) and \(A_{original} = 2\). Let \(A_{final}=x\times A_{original}\), then \(x=\frac{A_{final}}{A_{original}}=\frac{18}{2}=9\).

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 side - lengths are equal. For a dilation, the ratio of corresponding side - lengths is the scale factor. So the original and dilated rectangles are similar.

Answer:

(a) Area of original figure: \(2\) square units; Area of final figure: \(18\) square units.
(b) \(9\)
(c) False