The sides of a pentagon are represented by x , x+4, 2x+1, 2x+2, 3x. Determine each side if the perimeter is 52.

Since the pentagon is not regular, the lengths of the
sides are not equal.

The perimeter of any geometric shape
is the sum of the lengths of the sides of that shape.

We'll
note the perimeter as P and the sides as s1, s2...

Since
the pentagon has 5 sides, we'll calculate the perimeter
as:

P = s1 + s2 + s3 + s4 +
s5

We know, from enunciation, the value of the perimeter
and an expression for each side. We'll substitute all we know in the formula of
perimeter.

52 = x + (x+4) + (2x+1) + (2x+2) +
3x

We'll remove the brackets from the right
side:

52 = x + x + 4 + 2x + 1 + 2x + 2 +
3x

We'll combine like
terms:

52 = 9x + 7

We'll
subtract 52 both sides:

0 = 9x + 7 -
52

0 = 9x - 45

We'll add 45
both sides and we'll use symmetric property:

9x =
45

We'll divide by 9:

x =
5

 Now, we'll determine each
side:

s1 = x

s1
= 5

s2 = x+4

s2
= 5+4

s2 =
9

s3 = 2x +
1

s3 = 2*5 +
1

s3 = 
11

s4 = 2x +
2

s4 =
12

s5 =
3x

s5 =
15