Question

for the figure below, do a dilation centered at the origin with a scale factor of 4. 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

Explanation:

Step1: Find the area of the original figure

The original figure is a rectangle. From the graph, we can see that the length \(l = 2\) units and the width \(w=1\) unit.
The area formula of a rectangle is \(A = l\times w\).
So, \(A_{original}=2\times1 = 2\) square units.

Step2: Find the area of the dilated figure

When a figure is dilated with a scale factor \(k = 4\) centered at the origin, the length and width of the rectangle are multiplied by the scale factor.
The new length \(l'=2\times4 = 8\) units and the new width \(w'=1\times 4=4\) units.
Using the area formula \(A = l\times w\), we get \(A_{final}=8\times4=32\) square units.

Step3: Find the relationship between the areas

We know that \(A_{original} = 2\) and \(A_{final}=32\).
Let \(A_{final}=k\times A_{original}\), then \(k=\frac{A_{final}}{A_{original}}=\frac{32}{2}=16\).
Since \(k = 4\) (scale - factor for length), and the area scale - factor is \(k^{2}\) (because area is a two - dimensional measure, \(A = l\times w\), and both \(l\) and \(w\) are scaled by \(k\)), \(4^{2}=16\).

Step4: Determine if the figures are similar

Two figures are similar if the ratios of their corresponding side lengths are equal.
For two rectangles (original and dilated), the ratio of their corresponding side lengths is equal to the scale factor of dilation.
The original rectangle has side lengths \(a = 2\), \(b = 1\) and the dilated rectangle has side lengths \(a'=8\), \(b' = 4\).
\(\frac{a'}{a}=\frac{8}{2}=4\) and \(\frac{b'}{b}=\frac{4}{1}=4\). So, the original and dilated figures are similar.

Answer:

(a) Area of original figure: \(2\) square units; Area of final figure: \(32\) square units.
(b) Area of final figure \(=16\times\) Area of original figure.
(c) False.