Question

for the figure below, do a dilation centered at the origin with a scale factor of 4. 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 square with side length \(1\) unit. The area of a square is \(A = s^2\), where \(s\) is the side - length. So, \(A_{original}=1\times1 = 1\) square unit.

Step2: Find the side - length of the dilated figure

Since the scale factor \(k = 4\) and the dilation is centered at the origin, the side - length of the new square \(s_{new}=k\times s_{original}\). Here, \(s_{original} = 1\), so \(s_{new}=4\times1=4\) units.

Step3: Find the area of the dilated figure

Using the area formula for a square \(A = s^2\) again, with \(s = 4\), we get \(A_{final}=4\times4=16\) square units.

Step4: Relate the areas

We know that if the scale factor of a dilation is \(k\), the ratio of the areas of the dilated figure (\(A_{2}\)) to the original figure (\(A_{1}\)) is \(A_{2}=k^{2}A_{1}\). Here, \(k = 4\), so \(A_{final}=4^{2}\times A_{original}\)

Step5: Check for similarity

Two figures are similar if their corresponding angles are equal (for a square, all angles are \(90^{\circ}\)) and their corresponding sides are in proportion. After dilation, the ratio of the sides of the final figure to the original figure is equal to the scale factor. So, the original square and the dilated square are similar.

Answer:

(a) Area of original figure: \(1\) square unit; Area of final figure: \(16\) square units.
(b) Area of final figure \(= 16\times\) Area of original figure (since \(4^{2}=16\)).
(c) True.