Question

expressions
find the perimeter of the given shape. write your answer in the simplest form.
(the image shows a polygon with side lengths labeled as 3x+2, 3x+4, 4x+1, x, 3x, 2x-1 and another side which is equal in length to 3x (by the property of the shapes sides, inferred from the diagrams structure))

Explanation:

Step1: Identify all sides

The sides of the shape are \(3x + 2\), \(3x + 4\), \(4x + 1\), \(x\), \(3x\), \(2x - 1\), and two other sides (we need to check the horizontal and vertical lengths to find the missing sides). Wait, actually, to find the perimeter, we can use the method of translating the sides to form a rectangle (by moving the indented parts) or sum all outer sides. Let's list all the outer sides:

First, let's find the total horizontal length: The top - bottom horizontal sides. Wait, maybe a better way is to sum all the side lengths. Let's list each side:

1. \(3x + 2\)
2. \(3x + 4\)
3. \(4x + 1\)
4. \(x\) (the small horizontal side at the bottom right of the indent)
5. \(3x\) (the vertical side of the indent)
6. \(2x - 1\) (the horizontal side of the indent)
7. Now, the missing vertical side: The vertical side on the left of the indent. The total vertical length on the left: The left side is \(3x + 2\), and the indent's vertical side is \(3x\), so the missing vertical side (the one parallel to \(3x\)) should be \(3x + 2-(3x - (2x - 1))\)? Wait, no, maybe a better approach is to use the fact that in a polygon, the perimeter is the sum of all its sides. Let's re - examine the shape:

Looking at the vertical sides:
- The leftmost vertical side: \(3x + 2\)
- The rightmost vertical side: \(4x + 1\)
- The vertical side of the indent: \(3x\)
- The other vertical side (complementary to the indent) should be \(3x + 2-3x+(2x - 1)\)? No, this is getting complicated. Let's instead "unfold" the shape. If we move the indented part, we can see that the horizontal lengths:

The top horizontal length: \(3x + 4\)

The bottom horizontal length: \(x + 3x+(3x + 2-(2x - 1))\)? No, let's list all the sides as they are:

Wait, the correct way is to list all the sides:

Side 1: \(3x + 2\) (left vertical)

Side 2: \(3x + 4\) (top slant? No, looking at the diagram, it's a polygon with sides: \(3x + 2\) (left vertical), then a horizontal side (let's say length \(A\)), then a slant side \(3x + 4\), then a vertical side \(4x + 1\) (right vertical), then a horizontal side \(x\) (bottom right horizontal), then a vertical side \(3x\) (right vertical of indent), then a horizontal side \(2x - 1\) (bottom left horizontal of indent), then a vertical side (left vertical of indent) which is equal to \(3x + 2-3x=2\)? No, this is wrong.

Wait, let's use the standard method for such "indented" polygons: The perimeter is equal to the perimeter of the rectangle that would be formed if there were no indent plus twice the length of the indent's horizontal or vertical sides (depending on the indent).

The rectangle without indent would have length \(L=(3x + 4)+x + 2x - 1=6x + 3\) and width \(W = 4x + 1\) (if we consider the right vertical side) or \(W=3x + 2\) (left vertical side). Wait, no, the correct way is:

The horizontal sides:
- Top: \(3x + 4\)
- Bottom: \(x+3x + 2x - 1=6x - 1\)? No, this is incorrect.

Wait, let's sum all the sides as given in the diagram:

The sides are: \(3x + 2\), \(3x + 4\), \(4x + 1\), \(x\), \(3x\), \(2x - 1\), and two more sides. Wait, the missing sides:

The horizontal side between \(3x + 2\) and the indent: Let's call it \(S_1\)

The vertical side between \(2x - 1\) and \(3x + 2\): Let's call it \(S_2\)

From the vertical direction: The total vertical length on the left is \(3x + 2\), and the vertical length of the indent is \(3x\), so \(S_2=3x + 2 - 3x=2\)? No, and from the horizontal direction: The total horizontal length of the bottom (excluding the indent) should be equal to the top horizontal length. The top horizontal length is \(3x + 4\), and the bottom horizontal length is \(x + 3x+S_1\), and also \(S_1=3x + 4-(2x - 1)\)? No, this is too confusing. Let's instead list all the sides as per the diagram:

Looking at the diagram, the sides are:

1. \(3x + 2\) (left vertical)
2. A horizontal side (let's calculate its length: The top side is \(3x + 4\), and the indent's horizontal side is \(2x - 1\), so the horizontal side between \(3x + 2\) and the indent is \(3x + 4-(2x - 1)=x + 5\)? No, this is wrong.

Wait, the correct approach is to sum all the sides:

Perimeter \(P=(3x + 2)+(3x + 4)+(4x + 1)+x + 3x+(2x - 1)+(3x + 2)+(4x + 1 - 3x)\)

No, this is incorrect. Let's start over.

Let's identify each side:

- Left vertical: \(3x + 2\)
- Top: \(3x + 4\)
- Right vertical: \(4x + 1\)
- Bottom right horizontal: \(x\)
- Bottom right vertical (indent): \(3x\)
- Bottom left horizontal (indent): \(2x - 1\)
- Bottom left vertical (indent complement): \(3x + 2-3x = 2\) (since the left vertical is \(3x + 2\) and the indent's vertical is \(3x\), the remaining vertical length is \(2\))
- Bottom left horizontal (complement to top): The top horizontal is \(3x + 4\), and the indent's horizontal is \(2x - 1\) and the bottom right horizontal is \(x\), so the bottom left horizontal (complement) is \(3x + 4-(2x - 1)-x=3x + 4 - 2x + 1 - x = 5\)

Now sum all sides:

\((3x + 2)+(3x + 4)+(4x + 1)+x + 3x+(2x - 1)+2 + 5\)

Now combine like terms:

For \(x\) terms: \(3x+3x + 4x+x + 3x+2x=(3 + 3+4 + 1+3 + 2)x = 16x\)

For constant terms: \(2 + 4+1-1 + 2+5=(2 + 4)+(1-1)+(2 + 5)=6 + 0+7 = 13\)

Wait, this can't be right. Let's do it again.

Wait, the correct way is to realize that when we have an indented polygon, the perimeter is equal to the perimeter of the outer rectangle plus twice the length of the indent's horizontal or vertical sides.

The outer rectangle (if there were no indent) would have length \(L = 3x + 4+x=4x + 4\) and width \(W=4x + 1\) (vertical side) and the left vertical side is \(3x + 2\). Wait, no, the indent has a horizontal length of \(2x - 1\) and vertical length of \(3x\).

Wait, let's list all the sides properly:

1. Left vertical: \(3x + 2\)
2. Top horizontal: \(3x + 4\)
3. Right vertical: \(4x + 1\)
4. Bottom right horizontal: \(x\)
5. Bottom right vertical (indent): \(3x\)
6. Bottom left horizontal (indent): \(2x - 1\)
7. Bottom left vertical (indent complement): \(3x + 2-3x=2\) (because the left vertical is \(3x + 2\) and the indent's vertical is \(3x\), the difference is \(2\))
8. Bottom left horizontal (complement to top): The top horizontal is \(3x + 4\), and the indent's horizontal is \(2x - 1\) and the bottom right horizontal is \(x\), so this length is \(3x + 4-(2x - 1)-x=3x + 4 - 2x + 1 - x = 5\)

Now sum all sides:

\(3x + 2+3x + 4+4x + 1+x + 3x+2x - 1+2 + 5\)

Combine like terms:

- \(x\) terms: \(3x+3x + 4x+x + 3x+2x=(3 + 3+4 + 1+3 + 2)x = 16x\)
- Constant terms: \(2 + 4+1-1 + 2+5=13\)

Wait, but this seems off. Let's try another approach. Let's use the fact that the perimeter of a polygon is the sum of all its sides. Let's list all the sides as per the diagram:

Looking at the diagram, the sides are:

- \(3x + 2\) (left)
- A horizontal side (let's say length \(a\))
- \(3x + 4\) (top)
- \(4x + 1\) (right)
- \(x\) (bottom right horizontal)
- \(3x\) (right vertical of indent)
- \(2x - 1\) (bottom left horizontal of indent)
- \(b\) (left vertical of indent)

Now, from the vertical direction: \(b + 3x=3x + 2\) (since the leftmost vertical side is \(3x + 2\) and the two vertical sides of the indent sum up to it), so \(b = 2\)

From the horizontal direction: \(a+3x + 4=x + 2x - 1 + a\)? No, \(a=3x + 4-(x + 2x - 1)=3x + 4 - 3x + 1 = 5\)

Now sum all sides: \(3x + 2+5+3x + 4+4x + 1+x + 3x+2x - 1+2\)

Wait, \(3x + 2\) (left) + \(5\) (horizontal) + \(3x + 4\) (top) + \(4x + 1\) (right) + \(x\) (bottom right horizontal) + \(3x\) (indent vertical) + \(2x - 1\) (indent horizontal) + \(2\) (indent left vertical)

Now sum the \(x\) terms: \(3x+3x + 4x+x + 3x+2x=16x\)

Sum the constants: \(2 + 5+4+1-1 + 2=13\)

So the perimeter is \(16x + 13\)? Wait, no, let's check again.

Wait, maybe I made a mistake in identifying the sides. Let's use the correct method for a polygon with an indent: The perimeter is equal to the perimeter of the outer rectangle plus twice the length of the indent's horizontal or vertical sides.

The outer rectangle (if we ignore the indent) would have length \(L = 3x + 4\) and width \(W=4x + 1\), but no, the left side is \(3x + 2\). Wait, the vertical sides: the left side is \(3x + 2\), the right side is \(4x + 1\), the horizontal sides: the top is \(3x + 4\), and the bottom (without indent) would be \(3x + 4\). But we have an indent with horizontal length \(2x - 1\) and vertical length \(3x\). So the perimeter is \(2\times(3x + 4+(4x + 1))+2\times(2x - 1)\)? No, that's not right.

Wait, let's start over. Let's list all the sides as they appear in the diagram:

1. Left vertical: \(3x + 2\)
2. Top horizontal: \(3x + 4\)
3. Right vertical: \(4x + 1\)
4. Bottom right horizontal: \(x\)
5. Bottom right vertical (indent): \(3x\)
6. Bottom left horizontal (indent): \(2x - 1\)
7. Bottom left vertical (indent): Let's call it \(y\). Since the left vertical is \(3x + 2\), and the indent's vertical is \(3x\), then \(y=3x + 2 - 3x = 2\)
8. Bottom left horizontal (complement to top): The top horizontal is \(3x + 4\), and the bottom right horizontal is \(x\) and the indent's horizontal is \(2x - 1\), so this length is \(3x + 4-(x + 2x - 1)=3x + 4 - 3x + 1 = 5\)

Now sum all these sides:

\((3x + 2)+(3x + 4)+(4x + 1)+x + 3x+(2x - 1)+2 + 5\)

Now combine like terms:

- For \(x\): \(3x+3x + 4x+x + 3x+2x=(3 + 3+4 + 1+3 + 2)x = 16x\)
- For constants: \(2 + 4+1-1 + 2+5 = 13\)

Wait, but let's check with a different method. Let's add all the sides:

\(3x + 2+3x + 4=6x + 6\)

\(6x + 6+4x + 1 = 10x + 7\)

\(10x + 7+x=11x + 7\)

\(11x + 7+3x=14x + 7\)

\(14x + 7+2x - 1=16x + 6\)

\(16x + 6+2=16x + 8\)

\(16x + 8+5=16x + 13\)

Yes, so the perimeter is \(16x + 13\)

Step2: Combine like terms

After listing all the sides and adding them up, we combine the \(x\) terms and the constant terms. The \(x\) terms are \(3x+3x + 4x+x + 3x+2x = 16x\) and the constant terms are \(2 + 4+1-1 + 2+5 = 13\). So the perimeter is \(16x + 13\)

Answer:

The perimeter of the given shape is \(\boldsymbol{16x + 13}\)