The statement can be proven by using an indirect proof. Assume that the shortest distance from a point $P$ to a line $\ell$ is not the length of the perpendicular segment connecting the point to the line. Now suppose that $\overline{PB}$ is the shortest segment between point $P$ and line $\ell .$

By the Perpendicular Postulate, there is a segment from $P$ that is perpendicular to $\ell .$ Let that segment be $\overline{PA}.$

Notice that $△ABP$ is a right triangle. In this case, $\overline{BP}$ is the hypotenuse and, therefore, the longest side. In other words, $PA$ is less than $BP.$ The assumption that $\overline{BP}$ is the shortest segment, but not perpendicular, connecting point $P$ to line $\ell$ must be false because there exists a shorter segment, $\overline{PA},$ which is perpendicular to line $\ell .$ Therefore, the shortest distance from a point to a line is the length of the perpendicular segment connecting the point to the line.