Through each vertex of a triangle, a line parallel to the opposite side is drawn. The ratio of the perimeter of the new triangle, thus form…

Let us consider PQR be the original triangle

Given that through each vertex of a triangle, a line parallel to the opposite side is drawn

Let us consider DEF be the new triangle formed where DF side passes through vertex P of the original triangle

Similarly, DE side passes through vertex Q and EF side passes through vertex R

Here PR ? QE and PQ ? RE

∴ PQRE forms a parallelogram

Similarly PR ? DQ and QR ? PF

∴ PFQR forms a parallelogram

By the above two relations we can conclude that Q is the mid point of DE and PR is half of the DE

Because of the symmetry we can also say P is the mid point of DF and QR is half of the DF and R is the mid point of the side EF and PQ is half of the side EF

∴ we can conclude that each of the sides of the original triangle is half of the sides of the new triangle

Finally ratio of the perimeter of new triangle DEF to that of the original triangle PQR is 2 : 1