12. A square with side-length $10 \mathrm{~cm}$ long is drawn on a piece of paper. How many points on the paper are exactly $10 \mathrm{~cm}$ away from two of the vertices of this square?
A 4
B 6
C 8
D 10
E 12
Solution
For each of the four pairs of adjacent vertices, there is one point outside the square and one point inside the square that is exactly $10 \mathrm{~cm}$ from each vertex in the pair. Also each of the four vertices themselves is exactly $10 \mathrm{~cm}$ from the two adjacent vertices. Hence, there are 12 points in total that are exactly $10 \mathrm{~cm}$ from two of the vertices of the square and these points are all distinct. The diagram below shows the positions of these 12 points.