### Video Transcript

True or False: The shortest
distance from a point to a line equals the length of any line segment that
passes through the point and intersects with the line.

To answer this question, weโll
begin with a point and a line. Letโs the name the point ๐ถ and
the line ๐ด๐ต. The statement provided is
regarding the shortest distance from ๐ถ to line ๐ด๐ต. The claim is that any line
segment through the point to the line is the shortest.

So letโs sketch an arbitrary
line segment from the point to the line. Letโs name the point of
intersection with the line point ๐ท. If the given statement is true,
then we will not find any shorter distance from ๐ถ to line ๐ด๐ต. Letโs try sketching the
perpendicular distance from ๐ถ to the line. We label the point ๐ธ where the
perpendicular segment meets line ๐ด๐ต at a right angle.

We notice that we have formed a
right triangle: triangle ๐ถ๐ธ๐ท. Line segment ๐ถ๐ท is the
hypotenuse of this right triangle. We recall that the hypotenuse
of a right triangle is always the longest side. Therefore, ๐ถ๐ท is greater than
๐ถ๐ธ. This means that the given
statement is false. Any nonperpendicular line
segment from point ๐ถ to line ๐ด๐ต would be considered a hypotenuse and thus be
longer than the perpendicular distance.

We conclude that the
perpendicular distance between a point and a line is the shortest distance.