Question

A piece of thread 120 cm in length is to be cut in to 2 parts 1 part to form a rectangle and other part to form an equilateral triangle the length of the rectangle needs to be three less than twice its width ans each side of the triangle needs to be twice the length of the rectangle what is the length of each part of the thread

Answer 1

Let the width of the rectangle be xcm
Then the length of the rectangle = 2x - 3
Then the length of the triangle = 2(2x - 3) = 4x - 6
Perimeter of the rectangle = 2(l+w)
⇒2(2x - 3 + x)
⇒2(3x - 3)
⇒6x - 6
Perimeter of the equilateral triangle = 3a
⇒3(4x - 6)
⇒12x - 18
The perimeter of the rectangle + perimeter of the triangle = 120cm
⇒6x - 6 + 12x - 18 =120
⇒18x - 24 = 120
⇒18x = 120 + 24
⇒18x = 144
⇒x = 144/18
∴x = 8cm
One part = perimeter of the rectangle
⇒6x - 6
⇒6* 8 - 6
⇒48 - 6
⇒42cm
Another part = perimeter of the triangle
⇒12x - 18
⇒12* 8 - 18
⇒96 - 18
⇒78cm
∴The two parts are of the length 42cm and 78cm

Answer 2

Let the two parts be  X cm  and  120  -  X cm.

Let X cm form a rectangle of width W and length L. 
         2 W + 2 L = X cm
           L = 2 W - 3
         => L +3  + 2L = X     
         =>  3 L + 3 = X  -- -(1)

triangle perimeter:  3 a = 120 - X
             given  a = 2 L
             => 3 * 2L = 120 - X
               =>  6 L = 120 - X    --- (2)

  solve equations,    2X - 6 = 120 -X
             => X = 42 cm
  parts are:  42 cm and  78 cm