Question

At a certain time of the day a tree casts its shadow 12.5 feet long. If the height of the tree is 5 feet, find the height of another tree that casts its shadow 20 feet long at the same time.

Answer 1

The basic proportionality theorem states that if a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

In the above-shown figure, $DE$ is the height of one tree, $AB$ is the height of another tree. $AB\perp AC$ and $DE\perp AC$, therefore, $AB||DE$

As shown in the above figure, let $AB=x$, $CD=12.5$ ft, $CA=20$ ft and $DE=5$ ft.

Using the basic proportionality theorem, we have

$\frac{CD}{CA}=\frac{DE}{AB}$

$\Rightarrow \frac{12.5}{20}=\frac{5}{x}$

$\Rightarrow 12.5x=20\times 5$

$\Rightarrow 12.5x=100$

$\Rightarrow x=\frac{100}{12.5}=8$

Hence, the height of another tree is $8$ feet.