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.