11. In the diagram, the small equilateral triangles have area $4 \mathrm{~cm}^{2}$. What is the area of the shaded region?
A $80 \mathrm{~cm}^{2}$
B $90 \mathrm{~cm}^{2}$
C $100 \mathrm{~cm}^{2}$
D $110 \mathrm{~cm}^{2} \mathrm{E} 120 \mathrm{~cm}^{2}$
11. B Let $b \mathrm{~cm}$ be the length of the base and let $h \mathrm{~cm}$ be the height of the small equilateral triangles. The area of each triangle is $4 \mathrm{~cm}^{2}$ so $\frac{1}{2} \times b \times h=4$.
The shaded area is a trapezium with parallel sides of length $4 h$ and $5 h$ and with distance $\frac{5}{2} b$ between the parallel lines. Using the formula for the area of a trapezium, the shaded area is equal to $\frac{1}{2}(4 h+5 h) \times \frac{5}{2} b=\frac{1}{4} \times 45 b h$. From above, $b h=8$ so this area is equal to $\frac{1}{4} \times 45 \times 8 \mathrm{~cm}^{2}=90 \mathrm{~cm}^{2}$.
(Alternatively, one could observe that the first four horizontal rows of the shaded region have area equivalent to five of the small equilateral triangles while the fifth layer has area equivalent to half of that or 2.5 equilateral triangles. Hence the shaded region has area equivalent to 22.5 small equilateral triangles and so has area $22.5 \times 4 \mathrm{~cm}^{2}=90 \mathrm{~cm}^{2}$.)