How (m^2)/(n^2)=(m/n)^2 is Fundamental to Geometry!

OVERVIEW
The Common Core stresses the importance of students developing a deeper understanding of fundamental concepts and to discover/uncover the interrelatedness of mathematics. The discussion below can be used to demonstrate how a basic law of exponents is tied to the geometry of similar figures.

THE PROBLEM/INVESTIGATION
1) If the sides of 2 squares are in the ratio 2:1, show that their areas are in the ratio 4:1
(a) visually
(b) numerically by examining particular cases
(b)  algebraically

2) If the sides are in the ratio 3:1, do you think the areas will be in the ratio 6:1 or 9:1? Now do parts a-c as in 1).

3) If the ratio of the sides is 3:2 show algebraically that the ratio of the areas is 9:4.

4) Show algebraically that if the ratio of the sides of 2 squares is m:n then the ratio of their areas is (m/n)^2.

Note: How does this result connect to the idea that the area of a square varies directly as the square of its side length?

4) If squares are replaced by circles using radii or diameters in place of "sides" show that the results of questions 1-4 are the same.

How does this result connect to the idea that the area of a circle varies directly as the square of its radius or diameter (or circumference)?

REFLECTIONS
• Squares and circles are of course special cases of similar figures. Beyond this investigation lies the BIG IDEA:

The areas of 2-dim similar figures are proportional to the squares of their linear dimensions.

Note: In 3 dim, we can replace 'area' by what?

• Do you see this as one of the fundamental theorems of Euclidean geometry? Is it sufficiently stressed in textbooks and in the standards? Of course you may not feel as I do about all this!

• So what is the geometry connection to
(m/n)^3 = (m^3)/(n^3)...

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