What is the width of the street?

Question

There are two houses with a street in between them. two ladders of length $a$ and $b$ are standing across the street with their bases on the street touching the wall of the houses and their heads touching the wall of the house on the other side of the street. The two ladders intersect at a height $h$ above the ground. What is the width of the street in terms of a, b and h?

Answer 1

If $h_a,h_b$ are the heights of the ladder heads and $x$ is the width of the street, then $$x^2+h_a^2 =a^2,\quad x^2+h_b^2=b^2$$ and $$ x = \frac {h}{h_a}x+\frac{h}{h_b}x.$$ From the latter, $$ h_ah_b=h(h_a+h_b)$$