A narrow street is lined with tall buildings. An $x$ foot long ladder is rested at the base of the building on the right side of the street and leans on the building on the left side. A $y$ foot long ladder is rested at the base of the building on the left side of the street and leans on the building on the right side. The point where the two ladders cross is exactly $c$ feet from the ground. How wide is the street?
Best Answer
Let $a$ be the distance from the top of ladder $x$ to the ground and $b$ be the distance from the top of ladder $y$ to the ground. The projection on the ground of the point where the ladders cross divides the width $w$ of the street into two parts $\alpha$ (on the left) and $\beta$ (on the right). By similitude of triangles you have $$ {c\over a}={\beta\over w}\quad\hbox{and}\quad {c\over b}={\alpha\over w}. $$ Adding these two equalities leads to $$ {c\over a}+{c\over b}={\alpha+\beta\over w}=1. $$ It is now possible to express $a$ and $b$ in terms of $w$, $x$ and $y$ (via Pythagora's theorem) and thus obtain an equation for $w$.