15. A vertical stained glass square window of area $81 \mathrm{~cm}^{2}$ is made out of six triangles of equal area (see figure). A fly is sitting on the exact spot where the six triangles meet. How far from the bottom of the window is the fly sitting?
A $3 \mathrm{~cm}$
B $5 \mathrm{~cm}$
C $5.5 \mathrm{~cm}$
D $6 \mathrm{~cm}$
E $7.5 \mathrm{~cm}$
Solution
D

Let $h$ be the height of the fly above the base of the window. Each side-length of the square window of area $81 \mathrm{~cm}^{2}$ is $9 \mathrm{~cm}$. The two triangles that form the bottom part of the window have total area equal to a third of the whole window, namely $27 \mathrm{~cm}^{2}$. Hence $\frac{1}{2} \times 9 \times h=27$ so $h=27 \times 2 \div 9=6 \mathrm{~cm}$.