Find out if the triangles with vertices (2,6) , (1,1) (3,4) and (1,3), (4,8) and ( 2,4) are similar or not.

Triangles with vertices A(2,6) ,  B(1,1)  C(3,4) and
P(1,3), Q(4,8) and R( 2,4) to be verified whether similar or
not.

We find the lengths of the sides . We write them in
increasing orderof  the sides according to their lengths in each of the triangles. We
see whether there is the same  ratio between length of each side of the one to that of
another triangle:

AB^2 = (2-1)^2+(6-1)^2  = 1+25=  26. So
AB = sqrt 26.

AC^2 = (2-3)^2+(6-4)^2 = 1+4 = 5 , AC =
sqrt 5

BC^2 = (1-3)^2+(1-4)^2 = 4+9 = 13. BC =
sqrt13.

PQ^2 = (1-4)^2+(3-8)^2 = 9+25=34, PQ =
sqrt34

PR^2 = (1-2)^2+(3-4)^2 = 1+1 =2. PR = sqrt
2

QR^2 = (4-2)^2+(8-4)^2 = 4+16 = 20, QR =
sqrt20.

Therefore , sides of the rwo triangles in the
incresing orderare:

(sqrt5 , sqrt13 , sqrt26) and (sqrt2 ,
sqrt20 , sqrt34).

For similarity, the correspoding sides of
the triangles  should bear the  same ratio.

 sqrt5/sqrt2 
and  sqrt13/sqrt20  and  sqrt20/sqrt34 are all
different.

So the triangles are not
similar.