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Let's solve this step-by-step.

### Step 1: Determine \( x \) using the perimeter information

The problem states that the perimeter of hexagon \( ABCDEF \) is 40 cm. Adding up all given side lengths, we get:
\[
AB + BC + CD + DE + EF + FA = 40
\]
where:
- \( AB = 2x + 1 \)
- \( BC = 3x + 4 \)
- \( CD = 2x \)
- \( DE = 2x \)
- \( EF = 5x \)
- \( FA = 3x + 4 \)

Adding these together:
\[
(2x + 1) + (3x + 4) + (2x) + (2x) + (5x) + (3x + 4) = 40
\]

Simplify:
\[
17x + 9 = 40
\]

Subtract 9 from both sides:
\[
17x = 31
\]

Divide by 17:
\[
x = \frac{31}{17} = 1.82 \, \text{(approximately)}
\]

### Step 2: Substitute \( x \) to find side lengths

Now, substitute \( x = 1.82 \) into each expression to find the lengths of each side:
- \( AB = 2(1.82) + 1 = 4.64 \, \text{cm} \)
- \( BC = 3(1.82) + 4 = 9.46 \, \text{cm} \)
- \( CD = 2(1.82) = 3.64 \, \text{cm} \)
- \( DE = 2(1.82) = 3.64 \, \text{cm} \)
- \( EF = 5(1.82) = 9.1 \, \text{cm} \)
- \( FA = 3(1.82) + 4 = 9.46 \, \text{cm} \)

### Step 3: Calculate the area

To find the area of this irregular hexagon, we can split it into two rectangles:
1. Rectangle 1: Formed by sides \( AB \) and \( AF \).
   - Length = \( 4.64 \, \text{cm} \)
   - Width = \( 9.1 \, \text{cm} \)
   - Area = \( 4.64 \times 9.1 = 42.22 \, \text{cm}^2 \)

2. Rectangle 2: Formed by sides \( CD \) and \( DE \).
   - Length

The perimeter of the hexagon ABCDEF is 40 cm. Find the area of the hexagon given the side lengths: AB = (2x + 1) cm, BC = (3x + 4) cm, CD = 2x cm, DE = 2x cm, EF = 5x cm, FA = (3x + 4) cm.

Solve the problem of finding the area of a hexagon with given side lengths involving algebraic expressions. Learn how to calculate the perimeter and use it to find the value of x, then compute the area by splitting the shape into rectangles. Suitable for students in Grades 8-10.

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