21. The diagram shows two congruent equilateral triangles whose overlap is a hexagon. The areas of the smaller triangles, which are also equilateral, are $1,1,9,9,16$ and 16 , as shown.
What is the area of the inner hexagon?
A 68
B 58
C 48
D 38
E 28
21. D The ratio of the areas of the three smaller equilateral triangles which have vertices at $N, Q$ and $R$ respectively is $1: 9: 16$. So the ratio of the side-lengths of these three triangles is $\sqrt{1}: \sqrt{9}: \sqrt{16}=1: 3: 4$.
Hence the ratio of the side-length of triangle $L M N$ to that of triangle $P Q R$ is $1: 8$ and the ratio of the areas of these two triangles is $1^{2}: 8^{2}=1: 64$.
Therefore, as triangle $L M N$ has area 1, triangle $P Q R$ has area 64 .
So the area of the inner hexagon is $64-(1+9+16)=38$.