22. A caterpillar crawled up a smooth slope from $A$ to $B$, and crept down the stairs from $B$ to $C$. What is the ratio of the distance the caterpillar travelled from $B$ to $C$ to the distance it travelled from $A$ to $B$ ?
A $1: 1$
B $2: 1$
C $3: 1$
D $\sqrt{2}: 1$
E $\sqrt{3}: 1$
SolUtion
E

Let $h$ be the height of the slope. By dropping the perpendicular from $B$ to the base $A C$, we create two right-angled triangles $A B D$ and $B C D$. Angle $A B D=$ $(180-60-90)^{\circ}=30^{\circ}$ so triangle $A B D$ is half of an equilateral triangle and length $A B$ is twice $A D$. Let length $A D=x$ so that $A B=2 x$. Then by Pythagoras, $h^{2}=(2 x)^{2}-x^{2}=3 x^{2}$ so $h=\sqrt{3} x$. In triangle $B C D$ the angles are $90^{\circ}, 45^{\circ}, 45^{\circ}$ so $B C D$ is isosceles and the base $C D=B D=h$.
The total vertical height of the steps is equal to $B D=h$; the horizontal parts of the steps have total length equal to $C D=h$. The ratio of the distance travelled from $B$ to $C$ to the distance travelled from $A$ to $B$ is $2 h: 2 x=2 \sqrt{3} x: 2 x=\sqrt{3}: 1$.