4. An observer views the segment $A B$ from two points $C$ and $D$ which are $300 \mathrm{~m}$ apart, at angles of $30^{\circ}$. The lines $A D$ and $B C$ are mutually perpendicular. Determine the length of the segment $A B$.
Solution. Let $O$ be the intersection point of lines $A D$ and $B C$ (see diagram). The right triangles $A C O$ and $B O D$ are halves of equilateral triangles (they have internal angles of $30, 60^{\circ}$, and $90^{\circ}$). If we introduce the notations $O A=x$ and $O B=y$, we get $O C=x \sqrt{3}$

![](https://cdn.mathpix.com/cropped/2024_06_05_e0d95f5510204c7c0f0fg-4.jpg?height=418&width=488&top_left_y=1543&top_left_x=996)
and $O D=y \sqrt{3}$. Now, from the right triangle $C D O$ we have

$$
(x \sqrt{3})^{2}+(y \sqrt{3})^{2}=300
$$

from which we get

$$
x^{2}+y^{2}=30000
$$

Finally, from the right triangle $A O B$ we get

$$
A B=\sqrt{x^{2}+y^{2}}=\sqrt{30000}=100 \sqrt{3}
$$