Heights and distances by similarity

At a given hour the sun’s rays are parallel to each other, so they meet the ground everywhere at the same angle $\displaystyle \alpha$.
An upright stick 1 m long casts a shadow 1.6 m long, and a tree casts a shadow 12 m long:

• The stick and its shadow, and the tree and its shadow, make two right-angled triangles with the same acute angle $\displaystyle \alpha$, so the triangles are similar.
• The scale factor: $\displaystyle {k}={\frac{12}{1.6}}={7.5}$.
• The tree’s height: $\displaystyle {h}={7.5\cdot1}={7.5}$ m.

The object’s height $\displaystyle H$ over its shadow $\displaystyle S$ equals the stick’s height $\displaystyle h$ over its shadow $\displaystyle s$:

A building’s shadow is 18 m long.
At the same hour, the shadow of a man 1.8 m tall is 2.4 m long.
How tall is the building, in metres?

A flagpole 6 m tall casts a shadow 4.5 m long.
At the same hour, a tree’s shadow is 7.2 m long.
How tall is the tree, in metres?

On a cloudy day there is no shadow. A mirror $\displaystyle M$ is laid on the ground, and you step back from it until you see the top of the tree, $\displaystyle K$, in it:

• Light leaves the mirror at the same angle at which it arrived, so the angle $\displaystyle \alpha$ by the mirror is the same in both triangles.
• Both triangles also have a right angle – so they are similar.
• $\displaystyle {\frac{h}{1.6}}={\frac{15}{2}}={7.5}$, so $\displaystyle {h}={1.6\cdot7.5}={12}$ m.

Noa’s eyes are 1.5 m above the ground.
She stands 1.2 m from the mirror, and the mirror is 20 m from the building.
Noa sees the top of the building in the mirror.
How tall is the building, in metres?

The width $\displaystyle BC$ of a pond cannot be measured directly: the water is in the way. A point $\displaystyle A$ is marked on land, and $\displaystyle D$ and $\displaystyle E$ are marked on the lines $\displaystyle AB$ and $\displaystyle AC$, so that $\displaystyle DE\parallel BC$:

• The line $\displaystyle DE$ is parallel to $\displaystyle BC$, so the corresponding angles are equal, and the triangles $\displaystyle ADE$ and $\displaystyle ABC$ are similar.
• The side $\displaystyle AB$ is $\displaystyle AD+DB$, and the scale factor is $\displaystyle \frac{AB}{AD}$.

In the drawing: $\displaystyle {AD}={12}$ m, $\displaystyle {DB}={8}$ m, $\displaystyle {DE}={15}$ m.
How wide is the pond, $\displaystyle BC$, in metres?

A straight ramp rises from the ground.
2 m from the ramp’s start, measured along the ground, a support 0.6 m tall stands under it.
At the ramp’s end, 5 m from its start, stands another support.
How tall is the other support, in metres?