Construction of Perpendicular to a Line through a Point on it Video Lecture - Class 6

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let's talk about perpendiculars now we
all know that two line segments are rays
are say to be perpendicular if they
intersect such that the angles between
them are right angles so they the two
line segments should make 90 degrees to
each other so let us take some examples
so here you see these are two lines and
the angle between them and they are
intersecting at a point let's say this
point is point O and the angle that they
make with each other is 90 degrees so D
we can say that let's say if this is a b
c so in this case we can say that a B is
perpendicular on DC or we can say BC is
perpendicular on Abel let's take more
examples here also you see that there
are two lines which are intersecting and
also they are intersecting at 90 degrees
so therefore here also we can see that
the two red lines are perpendicular to
each other let us now learn how do we
draw perpendicular to a line through a
point on it so that means you will be
given a line you will have to draw a
perpendicular on that line to a
particular point on that line so here we
have two different ways to do this
construction one is using ruler and set
square the other is using ruler and
compass so first let us look at the
ruler set square construction let's say
this is the line given to you and you
need to draw a perpendicular at a point
P which is at 11 centimeters from this
end so let us call call this line as a B
so this a B is a line segment and let's
say that you have to draw a
perpendicular at some point P which is
at a distance of 11 centimeters from a
so you are given that p is at a distance
of 11
leaders from a so how will you locate me
so for that first put a ruler exactly
along the line in segment a B so here
make a note of eleven centimeter which
is somewhere here right so that means
this is our point P where we want to
draw a perpendicular now what we do
bring the set Square and place it
exactly above the ruler as shown in the
diagram now place it along the ruler and
gradually start shifting it towards
point B until you reach that point which
corresponds to 11 centimeters so now you
have shifted your set Square exactly at
point P now what you need to do now we
have already learned that in a set
square this angle is 90 degrees which
means that this line also is at 90
degrees with the base and the base line
is along the line segment evening so
this is pretty simple all you need to do
is just to draw a line along this
perpendicular of the set squares around
this side of the stretch square you just
draw a line now when you remove the set
square what do you see you see that this
angle is 90 degrees now if you want you
can verify the measure of this angle
using a protractor so in this fashion
you can draw perpendicular using a ruler
at the same square now let's see how do
we draw it with a ruler compass okay so
in this case again let's assume that
this is the line which is given to us
and where is point P so as I said that
point P is at a distance of 11
centimeter from point a so point P would
be somewhere here so this is our point P
now in this case what we do we take the
conkers we considering P as the center
now whichever point we consider as the
center we put the pointed end of the
compass on that point because wherever
you want to draw a circle you you will
not draw the circle on the
Center for draw the circle you need to
put the pointed in at the center and
then draw the circle with the pencil and
that is what you will do so in this case
also do the same thing we will consider
P as the center so pointed end will be
at P and with any convenient radius that
means this width of the compass could be
anything the angle between the two arms
could actually be any convenient
measurement so with any convenient
radius we draw an arc such that it
intersects the line at two points
something like this so you actually draw
an arc like this using the compass now
when you look at this are you see that
this arc actually intersects the line at
two points this is one point and this is
another point so let us name these
points so let us say this is point a and
this is point B so these are the two
points where the arc intersects the line
now what we do now we consider point B
as the center that means the pointed end
is at B and with the other end we draw
an arc now this time again we do not
have any fixed radius but we make sure
that this width of the compass or the
radius for which you are drawing the
circle that radius should be greater
than the half of this length that means
it should be greater than PB so the
radius with which you are drawing it
should be greater than PB similarly on
this side when you draw the radius
should be greater than April so
basically the length of a be half of
length of a be greater than that should
be the radius when you draw the circle
or draw the arc using this compass so
you will draw an arc like this
now remove the compass from here place
the compass on point a and repeat the
same thing so here you are considering a
as the center and B
a radius greater than ap you draw an arc
like this so basically if you draw the
circle you would get a circle like this
but you do not want that that circle
just the arc should be enough so once
you have drawn this you would see that
these two arcs they intersect each other
at a point now all you need to do is
join this point to point P and the
moment you join them you actually get
the perpendicular now aren't you curious
to know that what actually did we do why
did we do so much of complication now
what we were actually doing worse we
were making a 90 degrees with so much of
construction so now the next question
that might come to your mind is how are
we making a 90 degree like this now that
you will learn a little later when we
talk about drawing angles for now you
need to understand that perpendicular it
is that line which makes 90 degree to
the baseline so what we have actually
done here is we have done we have drawn
a circle here and now exactly now we
have made sure that point B is at the
center of a and B right that's because
it is a circle so ap is also a radius of
the circle PB is also radius of the
circle so obviously P is at the center
of a and B therefore when you actually
draw a circle without the same radius
from B and a circle from this of the
same radius from a so the point which is
the intersection of these two arcs that
points corresponds to perpendicular at
that point and point B lies exactly on
the straight line so on a straight line
actually means that this would be 90
degree to a B so in this fashion we draw
a perpendicular to a line using ruler
and compass if you want you recap the
slide again if you have any doubts or
any confusion but this construction is
of four step first step you draw an arc
with B as Center such that the arc
intersects the line at points a and B
step number two with B as the center
draw an arc such that the radius is
greater than
beebee step number three with BL center
draw and with air Center draw an arc
such that the radius is greater than ap
step number four join the point of
intersection of the two arcs to point P
and the line that you get is
perpendicular to a B so in this case if
let's call this as M so you can say that
PM is perpendicular to a thank you
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