Construction of a Triangle Similar to a given Triangle Video Lecture - Class 10

welcome to this session of geometry and
today we will look at how to construct a
similar triangle but before that let us
quickly revisit the concept of
similarity when it comes to a triangle
so if you have been given a triangle
when can we really say that any other
triangle is similar to this one we have
learnt in our previous session that it
happens whenever corresponding angles
between them are same and corresponding
sides are in the same ratio and as long
as these criterias are satisfied size of
the triangle won't matter it can be
bigger than the given triangle or
smaller or even an exact photocopy right
and also it won't make a difference if
we let's say rotate this similar
triangle in any of the direction because
these criterias will remain the same no
matter what but into a session we are
going to do something different we will
learn how to construct a similar
triangle when we do not have any
information about these criterias and
all we have been given is a triangle
with some dimension and we also know
something called as a scale factor now
what is the scale factor it is a ratio
that gives us an idea about how big or
how small a triangle has to be for
instance if we draw a line segment a-b
of length nine centimeter and we wish to
get another similar line segment but it
should be just two-third of this one
then what we can do is form three equal
portions on this line segment and just
consider length equal to two portions
that is three plus three equal to six
centimeters thus we say a B has been
scaled down to form CD by a scale factor
of 2 is 2 3 but did you notice when we
consider 2 equal portions the line
segment a B actually gets divided in a
ratio of 2 is to 1 so this is where we
have to be careful we should
not make any mistake while dealing with
these two kinds of ratios because scale
factor only indicates about the size
whereas the other one is only a dividing
ratio in the same way if we consider
triangle ABC then sides a B BC and AC
are just like line segments and if you
divide each of them in a ratio of 2 is
to 1 and use only these two portions
from each of the sides then we will be
able to construct a new triangle which
will be just 2/3 of this given triangle
and naturally it will be smaller in size
so this is the logic we will be using in
this session so for the construction of
a similar triangle we will have certain
requirements the first requirement will
be a compass which is the name for a
geometrical instrument do not confuse it
with the one we use for following
directions and we will also require any
kind of instrument that will just help
us to draw straight lines and it will
not have any markings on it for example
a straightedge so now let us go step by
step and understand all the steps of
construction first of all we have to
consider any side of the triangle to be
divided first just for the convenience
sake I am taking the side BC but we
cannot directly start dividing the side
BC in three equal parts because remember
we only have to use the compass and not
a measuring scale so in such a case we
cannot take a risk of dividing BC
inaccurately and since geometrical
construction is all about drawing
accurately so we'll have to take
appropriate measures so instead of
directly initiating construction on a
triangle we will draw array and this
will be our step number one draw away
from any of the vertex of the triangle
so let us say in this case I draw it
from vertex B and on this rate we will
start the construction we intend to do
now please note this ray has to be at a
shorter distance from the side BC which
means the angle between them should be
small now this is a crucial point to
remember because later will try to
connect this
vx with the side BC by some lines and
that proves to be easy and convenient
when the Ray is closer to the side BC
when this step is done we have to place
the pointed arm of the compass on the
vertex B and cut an arc of appropriate
length on the Ray BX and label this arc
as b1 now this length on the compass can
be of any measure as per our convenience
okay so whatever measurement you choose
maintain the same on your compass and
place it now on b1 and now cut an arc
and label it as B to repeat the same by
placing the compass on b2 this time and
make a new arc named B 3 so by the end
of this what we will get is the length B
b1 will be equal to b1 b2 and that will
be equal to b2 b3 now step 3 is to join
b3 with the vertex C so we will get a
line segment B 3 C and since we want to
isolate two equal parts we should even
draw a similar line segment from the
point B 2 now you guys tell me can I
draw it like this or like in any other
direction no it won't be correct because
that way we can see the side BC of the
triangle it doesn't really get divided
in a ratio of 2 is to 1 that we intend
to divide it into right so we have to be
cautious particularly while drawing this
line segment from the point b 2 it
should be done in such a way that it
proves to be parallel to B 3 C and for
that all we have to look into is the
angle between the Ray BX and B 3 C and
maintain the same angle over the point B
2 now it would be really easy to do that
with the help of a protractor or any
other instrument that readily measures
an angle right but yet again I should
remind you we just have to use a compass
and nothing else at all even for this
purpose and that's where lies our real
challenge and the trick so let me show
you how
we can do it draw an arc of appropriate
length with the compass from point B
three so that it cuts the Raby X and B 3
C repeat the same on point B two by
keeping the same measurement on the
compass now come back to the arc we made
earlier from B 3 and measure the length
of the arc from the point it touches BX
to the point it touches B 3 C now keep
the same measurement intact on the
compass cut another arc on the arc we
made from the point V 2 and now with the
help of a straight edge if we join B 2
with this arc intersection and then
further extend it to touch the side BC
say at Point D then not only will this
line segment B to D will be parallel to
B 3 C but also the side BC now has been
divided in a ratio of 2 is to 1 now we
can check this by a conventional way of
taking a scale with standard markings
and measuring the lengths so if let's
say BC was 12 centimeters we must get BD
as 8 centimeters and DC as 4 centimeters
because that is the 2 is to 1 ratio okay
or the second way to prove this is if
you recollect the well-known basic
proportionality theorem or BPT which
says if we pass a line through a
triangle in such a way that it is
parallel to one side of the triangle
then the other two sides will be divided
in the same proportions and if you
notice that is exactly what is happening
here if we consider B B 3 C as one
triangle then the line segment B to D
cuts this triangle in such a way that
it's parallel to the side B 3 C and if
you take the ratios of this divided
sides of the triangle what we will get
is a ratio of 2 is to 1 which means we
have successfully secluded two equal
parts on one side out of the three equal
parts now the last and the final thing
that remains after dividing the side BC
in a ratio of 2 is to 1 is to divide the
whole triangle in the same ratio
believe me this part is really easy all
we need to do is repeat step number four
on point D just like we did on point b2
and make a line segments parallel to the
side AC this time by maintaining the
angle between the sides BC and AC and
this way we will be able to divide even
the side a B in a ratio of 2 is to 1 and
let us call this new point as e on side
a B so can you notice this smaller
triangle EBD within the given triangle
ABC this is the similar triangle that we
were trying to obtain with the given
scale factor of 2 is 2 3 and just in
case if you would like to try to see if
really triangle abd is similar to
triangle ABC then you can make use of a
measuring scale and measure each side of
both the triangles I am sure you will
find the length of each side of triangle
abd to be 2/3 of the size of triangle
ABC and that's the beauty of this
construction even without really knowing
the angles and the measurements of sides
we could find a way to construct a
similar triangle accurately even then
with the help of instruments like
compass and straightedge
so please hit like if you have
understood how do we construct a similar
triangle and post any of your queries in
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