What percent of the area of the larger square is in the area of the smallest square?
Stuck again? Lets see what geogebra can do.
1) Draw four points A(0,7), B(7,7),C(7,0) and D(0,0).
2 Draw line segments to each of the points making a square.
3)Use the segment with given length from a point tool to make 4 segments from A,B,C,D of length 2. They will be labeled E,F,G,H.
4)Move these points until they are on the square as in the first picture.
5) Connect line segments to each of these 4 points to make the first interior square. See the second picture.
6)Use the 4 new points to make 4 new line segments of length 2 and put them on the interior square like in drawing 3.
7) You will have 4 new points I,J,K,L. Connect them with line segments to make the third square. See the third picture. Notice the length of one of the sides of the interior square.
The ratio of the two areas is:
Which is 31.6%
That is fairly close to the analytical answer,
Without any math or programming! The ability to make accurate diagrams is a powerful tool!
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.