Problem 2. Given a line $p$ with the equation $4 x+3 y-6=0$. What is the distance from the origin to this line?
Solution. The line intersects the coordinate axes at points $A(0,2)$ and $B\left(\frac{3}{2}, 0\right)$. The required distance is the length of the height dropped from vertex $O$ in the right triangle $ABO$, as shown in the diagram on the right. Let $\overline{ON}=x$. From the Pythagorean theorem, we have

$$
\overline{AB}=\sqrt{2^{2}+\left(\frac{3}{2}\right)^{2}}=\frac{5}{2}
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_b353751bdd87cc544500g-11.jpg?height=418&width=411&top_left_y=272&top_left_x=1054)

Now

$$
\begin{aligned}
& \frac{1}{2} \overline{OA} \cdot \overline{OB}=\frac{1}{2} \overline{AB} \cdot \overline{ON} \\
& \frac{1}{2} \cdot 2 \cdot \frac{3}{2}=\frac{1}{2} \cdot \frac{5}{2} x \\
& x=\frac{6}{5}
\end{aligned}
$$

Thus, the origin $O$ is $\frac{6}{5}$ units away from the line $p$.