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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, weโ€™ll begin with a point and a line. Letโ€™s the name the point ๐ถ and the line ๐ด๐ต. The statement provided is regarding the shortest distance from ๐ถ to line ๐ด๐ต. The claim is that any line segment through the point to the line is the shortest.

So letโ€™s sketch an arbitrary line segment from the point to the line. Letโ€™s name the point of intersection with the line point ๐ท. If the given statement is true, then we will not find any shorter distance from ๐ถ to line ๐ด๐ต. Letโ€™s try sketching the perpendicular distance from ๐ถ to the line. We label the point ๐ธ where the perpendicular segment meets line ๐ด๐ต at a right angle.

We notice that we have formed a right triangle: triangle ๐ถ๐ธ๐ท. Line segment ๐ถ๐ท is the hypotenuse of this right triangle. We recall that the hypotenuse of a right triangle is always the longest side. Therefore, ๐ถ๐ท is greater than ๐ถ๐ธ. This means that the given statement is false. Any nonperpendicular line segment from point ๐ถ to line ๐ด๐ต 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.

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