In this lesson we will learn about the method of finding the perpendicular distance
of a given point
from a given line L1: Y = a.X + b
There are several ways of finding the distance of a point from a line. I.e. using
co-ordinate geometry, linear algebra and simple trigonometry. In this lesson, we
will use trigonometric approach.
In the above diagram, Given line
and the given point is
Now let us look at the construction of the triangle formation in order to obtain the
perpendicular distance PC
We will first find the vertices of the triangle in order to get the side lengths and
then by applying Sine Rule
on triangle PAB
we will calculate the
desired distance PC
Draw a vertical line passing through the point P. This line L2:
will cut the
given line L1
at point A
). Similarly, draw a horizontal line passing
through the point P. This line L3:
will cut the given line L1
at point B
Now we need to calculate the vertices of the triangle PAB
Now we have three lines with following equations:
in L1 will give us the point A
and similarly plugging
in L1 will give us the point B
. Which can be calculated as:
Calculate the length of the sides AP,PB and AB of a triangle by the simple distance formula
in two-dimensional geometry. Distance formula :
formula we can calculate the side lengths AP,PB and AB.
Apply Sine rule
on common angle B
in triangle PAB
and triangle PBC
Now, Lets plug the distance formula for AP,BP and AB in the above expression
This is the final formula in terms of the given parameters xo, yo, a and b.
Hence PC is the desired length of a point P
from a line L1: Y = a.X + b
This Method uses the concept of linear algebra and some advance results on co-ordinate geometry.
The perpendicular length of a point
from a line a*X + b*Y + c=0
is given by the formula
Here the line is of the form of a*X + b*Y + c=0. We can always change the given
equation of line into this standard form and apply the above formula.
Also look at the solver based on the above concept.
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