Lesson The distance from a point to a straight line in a plane

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The distance from a point to a straight line in a plane


Let you are given a straight line and a point in a plane  (Figure 1).
What is the distance from the point to the straight line?

Let me remind you that the distance between any two different points in a plane is the length of the straight line segment connecting these points (see the lesson  Points and Straight Lines basics  under the topic  Points, lines, angles, perimeter  of the section  Geometry  in this site).

The  distance from a point in a plane to a straight line  in this plane is,  by the definition, the  length of the perpendicular drawn from the point to the straight line.

Why the perpendicular is chosen in this definition?  Why it is so remarkable or specific?

It is because the  perpendicular from the point to the straight line is the shortest segment among all the straight segments connecting the point with the straight line.

Theorem

The perpendicular from the point to the straight line is the shortest segment              
among all the straight segments connecting the point with the straight line.

Proof

The proof of this statement is very straightforward.
The  Figure 2  shows the straight line  RQ  in a plane, the point  P  in this plane
out of the line, the perpendicular  PQ  drawn from the point  P  to the line  QR,
and some other straight line segment  PR  connecting the point  P  with the straight
line  QR  at the point  R.  Let  d  be the length of the perpendicular  PQ 
and  t  be the length of the segment  PR.

Let us continue the perpendicular  PQ  in the same length into other half-plane
till the point  S  and connect the points  R  and  S  with the straigt line segment
RS.  Then the segments  PQ  and  QS  lie in one straight line  PS.  The triangles
PQR  and  SQR  are congruent as the right triangles with the common leg  QR


Figure 1.  The straight line in a plane,      
    the point  P  and the distance  d 
from the point  P  to the straight line


  Figure 2.  The straight line  QR,
the point  P,  the perpendicular  PQ
  and the straight line segment  PR 
and equal legs  PQ  and  SQ.  Hence, their hypotenuses  PR  and  SR  are of equal length:  |PR| = |SR| = t.

In the triangle  PRS  the sum of two sides  PR  and  SR  is longer than the third side  PS:  2%2At > 2%2Ad.  Hence,  t > d.   This is what has to be proved.

Summary

The perpendicular from the point to the straight line is the shortest segment among all the straight segments connecting the point with the straight line.
The distance from a point in a plane to a straight line in this plane is,  by the definition,  the length of the perpendicular drawn from the point to the straight line.


Note.  In a coordinate plane,  there is an explicit formula to calculate the distance from a point to a straight line.
See the lesson The distance from a point to a straight line in a coordinate plane  under the topic  Introduction to vectors, addition and scaling  of the section  Algebra-II  in this site.

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