# 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***, and***b***. Now, taking the conditions of the problem into effect, we can have these representations:***c***= longest side**

*c***= shortest side**

*b***= third side**

*a*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***, and***b***using the known formula of the perimeter.***c*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