Question Video: Determine the Shortest Distance from a Point to a Line Mathematics

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.

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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, 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.