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, we’ll
begin with a point and a line. Let’s the name the point 𝐶 and
the line 𝐴𝐵. The statement provided is
regarding the shortest distance from 𝐶 to line 𝐴𝐵. The claim is that any line
segment through the point to the line is the shortest.
So let’s sketch an arbitrary
line segment from the point to the line. Let’s name the point of
intersection with the line point 𝐷. If the given statement is true,
then we will not find any shorter distance from 𝐶 to line 𝐴𝐵. Let’s try sketching the
perpendicular distance from 𝐶 to the line. We label the point 𝐸 where the
perpendicular segment meets line 𝐴𝐵 at a right angle.
We notice that we have formed a
right triangle: triangle 𝐶𝐸𝐷. Line segment 𝐶𝐷 is the
hypotenuse of this right triangle. We recall that the hypotenuse
of a right triangle is always the longest side. Therefore, 𝐶𝐷 is greater than
𝐶𝐸. This means that the given
statement is false. Any nonperpendicular line
segment from point 𝐶 to line 𝐴𝐵 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.