The statement can be proven by using an . 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 , there is a segment from P that is perpendicular to ℓ. Let that segment be PA.
Notice that
△ABP is a . In this case,
BP is the 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.