21. Sophia has 52 identical isosceles right-angled triangles. She wants to make a square using some of them. How many different-sized squares could she make?
A 6
B 7
C 8
D 9
E 10
SolUtion
$\mathbf{C}$
Sophia can make a small square by joining two of her isosceles right-angled triangles, as shown in the first diagram. She can then join 4 or 9 or 16 or 25 of these small squares to make larger squares using up to 50 triangles. She can also create a different small square by joining four of her isosceles triangles, as shown in the second diagram. Note that the side-length of this square is $\sqrt{2}$ times the side-length of the previous small square. She can then join 4 or 9 of this second type of small square to make larger squares using up to 36 triangles. The eight squares described so far all have different side-lengths and it can be shown that there are no other possible sizes of squares. Hence Sophia can make 8 different-sized squares with the triangles she has.