An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$.
What is the area of triangle $ABC$?
TheIntegral wrote:
An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$.
What is the area of triangle $ABC$?
assuming the area of the triangle to be $n$, cutting the strips forms similar triangles with factor $\frac{2}{10}, \frac{3}{10}, \cdots, \frac{9}{10}$ and so on
this means the areas of the red strips are
$\frac{n}{100}, \frac{5n}{100}, \frac{9n}{100}, \frac{13n}{100}, \frac{17n}{100}$
and the areas of the blue strips are
$\frac{3n}{100}, \frac{7n}{100}, \frac{11n}{100}, \frac{15n}{100}, \frac{19n}{100}$
the difference between the two areas is $\frac{n}{10}=20$, so our area $n=\boxed{200}$.