How high is the kite?

To measure how high a kite flies without climbing anything, all you need is its string and a friend.

Find the kite's height $h$ without climbing.

Height $h$, no climbing.

Stand on the ground and let out fifty metres of string, pulled tight. Ask your friend to stand directly under the kite and measure how far away that spot is: forty metres.

Tight string $s=50$ m; distance from you to the spot under the kite $d=40$ m.

$s=50,d=40$.

The string, the ground and the line straight down from the kite make a right-angled triangle, and the string is its longest side. The square of fifty is two thousand five hundred, the square of forty is one thousand six hundred, and the difference is nine hundred. The square root of nine hundred is thirty.

String, ground and vertical make a right triangle with $s$ the longest side: $h^2+d^2=s^2$. So $h^2=50^2-40^2=2500-1600=900$, and $h=\sqrt{900}=30$ m.

$h^2+d^2=s^2\Rightarrow h=\sqrt{50^2-40^2}=\sqrt{900}=30$.

So the kite is thirty metres up, about ten floors of a building. The sides thirty, forty and fifty are three, four and five, each times ten. You never had to look up.

The kite is $30$ m up, about $10$ floors. The sides are $30:40:50=3:4:5$, times $10$.

$30:40:50=3:4:5$, and $h=30$ m.