Solve the following problem using the properties of similar triangles. A 40-foot flagpole outside an office building casts a 25-foot shadow. What is the height of the office building if the shadow it casts is 200 feet long?
We have two similar triangles: the triangle formed by the office building, its shadow, and the flagpole, and the triangle formed by the flagpole, its shadow, and the ground.
Let $h$ be the height of the office building. From the given information, we have $\frac{h}{40} = \frac{200}{25}$. Solving for $h$, we find $h = \frac{40}{25} \cdot 200 = \boxed{320}$ feet.
The sides of a triangular path measure 7 m, 9 m, and 11 m. Find the perimeter of a similar triangle in which the shortest side has a length of 21 meters.
Let $x$ be the length of the shortest side of the similar triangle.
We know that the ratio of corresponding sides in similar triangles is constant. Therefore, we can set up the proportion $\frac{7}{x} = \frac{11}{21}$.
Solving this proportion for $x$, we find $x=\frac{7 \cdot 21}{11} = \frac{147}{11} = 13.36$.
Therefore, the perimeter of the similar triangle is $7 + 9 + \frac{147}{11} = \boxed{27.36}$ meters.