Question

1. can similarity always/sometimes/never be deduced from comparing two different areas of the same shape? explain.
2. when comparing two similar figures, the scale factor that resized the first figure to the second is k. what scale factor resizes the area of the first figure to the area of the second figure? explain your rationale and why your answer does not depend on the shape of the figure.

Explanation:

Question 3

Similarity is about the shape (angles and side - length ratios). Area is a measure of space within the shape. Just because two areas of the same shape are different, we can't be sure of similarity. For example, a rectangle with length \(l_1 = 4\), width \(w_1=2\) (area \(A_1 = 8\)) and a rectangle with length \(l_2 = 6\), width \(w_2 = 3\) (area \(A_2=18\)) are similar (ratio of sides \(l_1:l_2=w_1:w_2 = 2:3\)). But a rectangle with \(l = 5\), \(w = 3\) (area \(A=15\)) and a rectangle with \(l = 9\), \(w = 2\) (area \(A = 18\)) are not similar (side - length ratios \(5:9
eq3:2\)).

Let's assume we have a two - dimensional shape. If the scale factor of the side - lengths of two similar figures is \(k\) (i.e., if the side - length of the first figure is \(s_1\) and the side - length of the second figure is \(s_2\), then \(\frac{s_2}{s_1}=k\)).
For a general two - dimensional shape, if we consider a small element (like a square for a more intuitive approach, but it works for any shape). The area of a square with side - length \(s_1\) is \(A_1=s_1^2\), and the area of a square with side - length \(s_2 = ks_1\) is \(A_2=(ks_1)^2=k^{2}s_1^{2}\). So, \(\frac{A_2}{A_1}=k^{2}\).
The formula for the area of any two - dimensional shape (using integration, \(\int\int dxdy\) in Cartesian coordinates or other coordinate systems) is based on the linear dimensions. When we scale the linear dimensions by a factor \(k\), the double - integral (which represents the area) will scale by \(k^{2}\) because both the \(x\) and \(y\) (or equivalent) directions are scaled by \(k\).

Answer:

Similarity can sometimes be deduced from comparing two different areas of the same shape.

Question 4