Use Cartesian principles to find the length of a line perpendicular to another?

Answer 1

length of perpendicular from (x1,y1) to the line Ax+By+C=0 is Ax1+By1+C/(A^2+B^2)^1/2

Answer 2

construction:

in the 1st quadrant of the
cartesian plane

1)draw line AB cutting x-axis at A
and y-axis at B
2)draw line OT(thro origin)
perpendicular to AB,cutting AB
at R.length OR=p,angle ROA=t
3)drop vertical line from point Q
(x1,y1) to cut x-axis at M
let d=the perpendicular distance from
Q(x1,y1) to AB(whose equation is
x1cost+y1sint =p)

then the sum of the projections of
OM and MQ on the line OT=the
projection of OQ on line OT
therefore,
x1cost+y1sint=p+d
>>>d=x1cost+y1sint-p
if a figure be drawn in which
O and Q are on the same side
as the line AB,it will be seen that
d= -x1cost-y1sint+p
hence we have the rule:if the
equation of the line be written in
the form -xcost-ysint+p=0, ie
with the constant term positive,
the perpendicular from a point
Q(x1,y1) is the value of
-x1cost-y1sint+p-if this expression
is positive,O and Q are on the same
side of the line,and if negative,
O and Q are on opposite sides of the line

any one form of the equation of a straight
line can be converted into any other
form,as a line has essentially only
one equation
if the equation of the line be
ax+by+c=0,this may be written
y=(-a/b)x-c/b and the line has a
gradient -a/b and makes the
intercept -c/b on the y-axis
the second form is the first
multiplied by 1/b
likewise,if ax+by+c=0 and
-xcost-ysint+p=0,in which c
and p are both positive,are
equations of the same line
cost/-a=sint/-b=p/c
=1/sqrt(a^2+b^2),
since cos^2(t)+sin^2(t)=1

the equation ax+by+c=0 may
therefore be written in the
'perpendicular' form by dividing by
sqrt(a^2+b^2),giving

ax/sqrt(a^2+b^2)
+by/sqrt(a^2+b^2)
+c/sqrt(a^2+b^2)=0

hence,the perpendicular distance of
the point (x1,y1) from the line
ax+by+c=0 is the numerical value of

(ax1+by1+c)/sqrt(a^2+b^2)

i hope that this helps

Answer 3

A line of the form y = mx + b has a line perpendicular to it of the form y = -(1/m)x + b which can be verified utilizing a Cartesian coordinate plane and the Pythagorean Thereom.

Answer 4

I`m going on holiday in 3 weeks so I don`t have time to read a question of this length!
A bit much really.

Answer 5

what information do you have to start with?

question seems a bit open ??

I would rotate so that the line is perpendicular to the x axis and then subtract the cartisian co-ordinates of the start and finish points?

Have I missed the point?