the shortest distance from a point to astraight line is?

Answer 1

Is a line perpendicular to it. The proof is easy.
Draw a line not perpendicular to the straight line.
Call this line L1. Now draw a line perpendicular to the straight
line. Call it L2. Note that you have a rt triangle with sides made of L1 and a side from the intersection of L1 and the original line
to the intersection of L2 and the original line and a hypotenuse which is L2. That makes L2 longer than L1 . So the shortest distance is necessarily perpendicular.

Answer 2

A straight line is a breadthless length that stretches forever in both directions.The shortest distance from a point to a line is the perpendicular. Let me explain better taking an example:
Take a line RS and a point P not lying on RS. The shortest distance between P and RS in the line segment connecting P to a point Q lying on RS such that:
Angle PQR (or) Angle PQS = 90 degrees

Answer 3

The equation of a line defined through two points P1 (x1,y1) and P2 (x2,y2) is

P = P1 + u (P2 - P1)
The point P3 (x3,y3) is closest to the line at the tangent to the line which passes through P3, that is, the dot product of the tangent and line is 0, thus

(P3 - P) dot (P2 - P1) = 0

Substituting the equation of the line gives


[P3 - P1 - u(P2 - P1)] dot (P2 - P1) = 0
Solving this gives the value of u

u = [(x3-x1)(x2-x1) + (y3-y1)(y2-y1)]/[IIp2-p1II^2]

[By II II^2 i mean the usual euclidean distance between the points]


Substituting this into the equation of the line gives the point of intersection (x,y) of the tangent as

x = x1 + u (x2 - x1)
y = y1 + u (y2 - y1)

The distance therefore between the point P3 and the line is the distance between (x,y) above and P3.

Answer 4

A straight line perpendicular to the origianl straight line.

Answer 5

A straight line!