Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
A man paved a square courtyard and then decided that it was too
small. He took up the tiles, bought 100 more and used them to pave
another square courtyard. How many tiles did he use altogether?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
The area of the inner square has area of 0.2 square units. First
draw the original square, then remove the triangles. The triangles
can then be added to the trapeziums to make four equal squares.
This gives a total of five equal squares which have a total area of
1 square unit. Therefore each of the smaller squares must have an
area of 0.2 square units.
If we don't use the midpoints on the sides of the big square the
problem can be generalised by considering the lengths as shown in
the diagram. In the above solution k = 1/2 and a
= b .
By enlargement (or similar triangles) c = ka
Using Pythagoras Theorem twice we get $a^2 + c^2 = k^2$ and also
$(a + b + c)^2 = 1 + k^2$
By some routine algebra we get $a^2 = k^2/(1 + k^2)$
and the area of the inner square is $b^2 = (1 - k)^2/(1 +