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The image shows a hexagon \( ABCDEF \) with different side lengths and some variables included. We're given:

- \( AB = 25 \, \text{cm} \)
- \( BC = (x + 2) \, \text{cm} \)
- \( CD = 8 \, \text{cm} \)
- \( EF = 7 \, \text{cm} \)
- \( AF = (x + 6) \, \text{cm} \)
- \( EF = 7 \, \text{cm} \)

We are also told that the area of the hexagon is \( 258 \, \text{cm}^2 \) and are asked to find the value of \( x \).

### Approach to solve for \( x \):

1. **Divide the hexagon into simpler shapes**: The hexagon can be divided into a rectangle \( ABCD \) and two right-angled triangles (one on the left and one on the right side).

2. **Area of the rectangle**: 
   The rectangle \( ABCD \) has dimensions:
   - Height = \( AB = 25 \, \text{cm} \)
   - Width = \( BC = (x + 2) \, \text{cm} \)

   So, the area of rectangle \( ABCD \) is:
   \[
   \text{Area of rectangle} = \text{Height} \times \text{Width} = 25 \times (x + 2)
   \]

3. **Area of triangle \( DEF \)**: 
   This is a right-angled triangle with:
   - Base \( EF = 7 \, \text{cm} \)
   - Height \( CD - EF = 8 - 7 = 1 \, \text{cm} \)
   
   The area of triangle \( DEF \) is:
   \[
   \text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 7 \times 1 = 3.5 \, \text{cm}^2
   \]

4. **Area of trapezium \( AFE \)**: 
   Using the trapezium \( AFE \), we can calculate the area based on given data, subtracting any unnecessary elements.

Let's calculate the value of \( x \).The value of \( x \) is approximately \( 8.18 \, \text{cm} \).

Would you like further details or clarification on the steps involved?

Here are 5 related questions for practice:
1. How would the formula for the area change if \( EF \) had a different length?
2. Can you calculate the perimeter of the hexagon for \( x = 8.18 \)?
3. How does the shape of the hexagon influence the calculation of its area?
4. What would happen to the total area if one side of the hexagon was increased by 5 cm?
5. How would you find the height of the hexagon if it was irregular?

**Tip**: For irregular polygons, dividing the shape into simpler geometric forms can make area calculations easier.

The diagram shows a hexagon ABCDEF with various side lengths given, and you are tasked with calculating the value of x based on the area of the hexagon being 258 cm².

Learn how to calculate the value of x in a hexagon with given side lengths and an area of 258 cm². This problem involves dividing the hexagon into simpler shapes and applying basic geometry formulas. Suitable for Grades 10-12.

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