Question

(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 similar.
true false

Explanation:

Step1: Calculate 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. Using the formula for the area of a rectangle \(A = l\times w\), we have \(A_{original}=2\times1 = 2\) square units.

Step2: Assume a dilation factor (since the dilation tool is not clear - but for a general dilation, if the dilation factor is \(k\)).

If we assume a dilation (for example, if we consider the properties of dilation. The area of a dilated figure \(A_{final}=k^{2}A_{original}\). If we assume a standard dilation (from the scale - if we consider a common case, say dilation factor \(k = 4\) (by counting grid - like if original is \(2\times1\) and final is \(8\times4\))). Then \(A_{final}=8\times4=32\) square units. And \(\frac{A_{final}}{A_{original}}=\frac{32}{2}=16\) (since \(k = 4,k^{2}=16\))

Step3: Check for similarity

Two figures are similar if their corresponding angles are equal (for rectangles, all angles are \(90^{\circ}\)) and their side - lengths are in proportion. For a dilated figure, the ratio of corresponding side - lengths is constant (the dilation factor). So the original and final figures (rectangles) are similar.

Answer:

(a) Area of original figure: \(2\) square units; Area of final figure: \(32\) square units (assuming a dilation factor \(k = 4\)).
(b) Area of final figure \(=16\times\) Area of original figure (assuming \(k = 4,k^{2}=16\)).
(c) True.