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The statement can be proven by using an indirect proof. Assume that the shortest distance from a point P to a line l is not the length of the perpendicular segment connecting the point to the line. Now suppose that PB is the shortest segment between point P and line l.
By the Perpendicular Postulate, there is a segment from P that is perpendicular to l. Let that segment be PA.
Notice that △ ABP is a right triangle. In this case, BP is the hypotenuse and, therefore, the longest side. In other words, PA is less than BP. PA < BP The assumption that BP is the shortest segment, but not perpendicular, connecting point P to line l must be false because there exists a shorter segment, PA, which is perpendicular to line l. Therefore, the shortest distance from a point to a line is the length of the perpendicular segment connecting the point to the line.