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 = × area of original figure (c) true or false? the original figure and the final figure are similar. true false

Explanation:

Step1: Find the area of the original figure

The original figure is a rectangle. The length of the original rectangle is \(8 - 4=4\) units and the width is \(6 - 4 = 2\) units.
Using the formula for the area of a rectangle \(A = l\times w\), we have \(A_{original}=4\times2=8\) square units.

Step2: Find the dimensions of the dilated figure

When a figure is dilated with a scale factor \(k=\frac{1}{2}\) centered at the origin, if the original length \(l\) and width \(w\), the new length \(l_{new}=k\times l\) and new width \(w_{new}=k\times w\).
The new length \(l_{new}=\frac{1}{2}\times4 = 2\) units and the new width \(w_{new}=\frac{1}{2}\times2=1\) unit.

Step3: Find the area of the dilated figure

Using the area formula \(A = l\times w\) for the dilated rectangle, \(A_{final}=2\times1 = 2\) square units.

Step4: Find the relationship between the areas

We want to find \(x\) such that \(A_{final}=x\times A_{original}\). Substituting \(A_{final} = 2\) and \(A_{original}=8\), we get \(2=x\times8\), so \(x=\frac{2}{8}=\frac{1}{4}\).

Step5: 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. Since we have a dilation (a transformation that changes the size but not the shape, and keeps the ratios of corresponding side - lengths equal), the original and dilated rectangles are similar.

Answer:

(a) Area of original figure: \(8\) square units; Area of final figure: \(2\) square units
(b) Area of final figure \(=\frac{1}{4}\times\) Area of original figure
(c) True