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In a triangle ABC, parallel lines are drawn to each side cutting the other two sides at the mid points. A new triangle is formed inside the original triangle. Find the ratio of the perimeter of the new triangle to perimeter of the ABC.

In a triangle ABC, parallel lines are drawn to each side cutting the other two sides at the mid points. A new triangle is formed inside the original triangle. Find the ratio of the perimeter of the new triangle to perimeter of the ABC. Correct Answer 1 : 2

Consider the following figure:

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Formula:

Perimeter of a triangle = (a + b + c)

Where

a = Side BC

b = Side CA

c = Side AB

Let the sides be AB = x; BC = y; AC = z

As shown in the figure, D, E and F are the mid points of sides AB, BC and AC respectively.

⇒ AD = DB = x/2

⇒ BE = EC = y/2

⇒ CF = FA = z/2

Now, DAFE form a parallelogram

⇒ DA = EF = x/2 and FA = DE = z/2      ....(1)

Also, DFEC form a parallelogram

⇒ DF = EC = y/2 and DE = CF = z/2      ....(2)

From (1) and (2)

EF = x/2; DF = y/2; DE = z/2      ....(3)

Now,

Perimeter of ∆ABC = x + y + z      ....(4)

Perimeter of ∆DEF = x/2 + y/2 + z/2 = (x + y + z)/2      ....(5)

From (4) and (5)

Ratio = Perimeter of ∆DEF/Perimeter of ∆ABC

⇒ (x + y + z)/2/(x + y + z) = 1/2  = 1 : 2

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