Question

for the figure below, do a dilation centered at the origin with a scale factor of 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 = × 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, we can see that the width \(w = 1\) (from \(x = 2\) to \(x = 3\)) and the height \(h=8\) (from \(y = 0\) to \(y = 8\)).
The area formula for a rectangle is \(A = w\times h\).
So, \(A_{original}=1\times8 = 8\) square units.

Step2: Find the area of the dilated figure

For a dilation centered at the origin with a scale factor \(k = 2\), the new width \(w'=k\times w\) and the new height \(h'=k\times h\).
So, \(w'=2\times1 = 2\) and \(h'=2\times8=16\).
Using the area formula \(A = w\times h\) for the dilated rectangle, \(A_{final}=2\times16=32\) square units.

Step3: Find the relationship between the areas

We want to find \(n\) such that \(A_{final}=n\times A_{original}\).
Substitute \(A_{final} = 32\) and \(A_{original}=8\) into the equation \(32=n\times8\).
Solving for \(n\), we get \(n=\frac{32}{8}=4\).

Step4: Determine if the figures are similar

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 the original rectangle with side - lengths \(a = 1\) and \(b = 8\), and the dilated rectangle with side - lengths \(a'=2\) and \(b' = 16\), \(\frac{a'}{a}=\frac{2}{1}=2\) and \(\frac{b'}{b}=\frac{16}{8}=2\).

Answer:

(a) Area of original figure: \(8\) square units; Area of final figure: \(32\) square units.
(b) \(4\).
(c) False.