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The longest side of a triangle is twice as long as the shortest side

The longest side of a triangle is twice as long as the shortest side and 2 cm longer than the third side. If the perimeter of the triangle is 33cm, what is the length of each side?

 
Solution:
 
Let us represent the 3 sides of the triangle as sides a, b, and c. Now, taking the conditions of the problem into effect, we can have these representations:
 
c    =  longest side
b    =  shortest side
a    = third side
P    = perimeter of the triangle
 
Thus, the conditions in the problem can be translated algebraically as:
 
c = (2)b                        [Eq. 1]
c = a + 2                      [Eq. 2]
 
Hence,
 
2b = a + 2                   [Eq. 3]
 
Now given the 2 equations, we can solve for the values of a, b, and c using the known formula of the perimeter.
 
P    =  a + b + c
33 = a + b + c
 
Substituting Eq. 1 in the perimeter equation above gives
 
33 = a + b + 2b
33 = a +3 b                      [Eq. 4]
 
Eq. 3 can be rewritten as follows:
2b =  a + 2
-2   =  a - 2b
2    = 2b - a                       [Eq. 5]
 
Now we can use Eq. 4 and Eq. 5 to solve the values of the variables a and b.
 
33 = a +3 b                      [Eq. 4]
2    = 2b - a                       [Eq. 5]
 
Gives
 
35 = 5b
35/5 = b
7    = b
 
Using Eq. 1 to solve for c:
 
c    = (2)b                         [Eq. 1]
c    = 2(7)
c    = 14
 
Using the perimeter formula to solve for a:
 
33 = a + b + c                  [Eq. 4]
33 = a +7 + 14
33 = a + 21
33 – 21      = a
12 = a
 
Therefore, the sides of the triangle measure 14, 12 and 7 and the perimeter is 33.
 
Check:
 
Perimeter
33 = a + b + c
33 = 12 + 7 +14
33 =  33