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

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<H2>The distance from a point to a straight line in a plane</H2>

Let you are given a straight line and a point in a plane &nbsp;(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 &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Points-and-Straight-Lines-basics.lesson>Points and Straight Lines basics</A>&nbsp; under the topic &nbsp;Points, lines, angles, perimeter&nbsp; of the section &nbsp;Geometry&nbsp; in this site).

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

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

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

<H3>Theorem</H3><TABLE>
 <TR>
 <TD>The perpendicular from the point to the straight line is the shortest segment&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

among all the straight segments connecting the point with the straight line.

Proof

The proof of this statement is very straightforward. 
The &nbsp;Figure 2&nbsp; shows the straight line &nbsp;RQ&nbsp; in a plane, the point &nbsp;P&nbsp; in this plane

out of the line, the perpendicular &nbsp;PQ&nbsp; drawn from the point &nbsp;P&nbsp; to the line &nbsp;QR,

and some other straight line segment &nbsp;PR&nbsp; connecting the point &nbsp;P&nbsp; with the straight

line &nbsp;QR&nbsp; at the point &nbsp;R. &nbsp;Let &nbsp;{{{d}}}&nbsp; be the length of the perpendicular &nbsp;PQ&nbsp;

and &nbsp;{{{t}}}&nbsp; be the length of the segment &nbsp;PR.

Let us continue the perpendicular &nbsp;PQ&nbsp; in the same length into other half-plane

till the point &nbsp;S&nbsp; and connect the points &nbsp;R&nbsp; and &nbsp;S&nbsp; with the straigt line segment

RS. &nbsp;Then the segments &nbsp;PQ&nbsp; and &nbsp;QS&nbsp; lie in one straight line &nbsp;PS. &nbsp;The triangles

PQR&nbsp; and &nbsp;SQR&nbsp; are congruent as the right triangles with the common leg &nbsp;QR

</TD>
 <TD>

{{{drawing( 252, 250, -2.0, 8.1, -4.0, 6.0,
            grid(1), 
            line (-3.6, -2.8,  8.0, 3.0),

circle (2, 5, 0.15),
            locate (2.1, 5.6, P),
            locate (3.1, 3.6, d),
        red(line(2, 5, 4, 1))
)}}}

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

</TD>
 <TD>

{{{drawing( 252, 250, -2.0, 8.1, -4.0, 6.0,
            grid(1), 
            line (-3.6, -2.8,  8.0, 3.0),

circle (2, 5, 0.15),
            locate (2.1, 5.6, P),
            locate (3.1, 3.6, d),
        red(line(2, 5, 4, 1)),
            locate (4.1, 1.75, Q),

blue(line (2.0, 5.0, 3.0, 0.5)),
            locate (2.5, 1.0, R),
            locate (2.1, 2.6, t),

red(line(4, 1, 6, -3)),
       blue(line (3.0, 0.5, 6, -3)),
            locate (6.1, -3.0, S)

)}}}

&nbsp;&nbsp;Figure 2. &nbsp;The straight line &nbsp;QR,
the point &nbsp;P, &nbsp;the perpendicular &nbsp;PQ
&nbsp;&nbsp;and the straight line segment &nbsp;PR&nbsp;

</TD>
 </TR>
</TABLE>and equal legs &nbsp;PQ&nbsp; and &nbsp;SQ. &nbsp;Hence, their hypotenuses &nbsp;PR&nbsp; and &nbsp;SR&nbsp; are of equal length: &nbsp;|PR| = |SR| = {{{t}}}.

In the triangle &nbsp;PRS&nbsp; the sum of two sides &nbsp;PR&nbsp; and &nbsp;SR&nbsp; is longer than the third side &nbsp;PS: &nbsp;{{{2*t}}} > {{{2*d}}}. &nbsp;Hence, &nbsp;{{{t}}} > {{{d}}}. &nbsp;&nbsp;This is what has to be proved.

<H3>Summary</H3>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, &nbsp;by the definition, &nbsp;the length of the perpendicular drawn from the point to the straight line.

Note. &nbsp;In a coordinate plane, &nbsp;there is an explicit formula to calculate the distance from a point to a straight line.

See the lesson <A HREF=http://www.algebra.com/algebra/homework/Vectors/The-distance-from-a-point-to-a-straight-line-in-a-coordinate-plane.lesson>The distance from a point to a straight line in a coordinate plane</A>&nbsp; under the topic &nbsp;Introduction to vectors, addition and scaling&nbsp; of the section &nbsp;Algebra-II&nbsp; in this site.