Let's break down the problem step by step and calculate the value of $x$.

Given Information:

• $BC = AB = 3x + 5$ meters
• $FE = 3x + 21$ meters
• $AF = DE = 3x + 2$ meters
• The perpendicular distance between $BC$ and $FE$ is $3x + 17$ meters.
• The perimeter of the hexagon $ABCDEF$ is 172 meters.

Approach:

The perimeter of the hexagon is the sum of all its side lengths. From the diagram and the given values, we can write the expression for the perimeter as:

$\text{Perimeter} = AB + BC + CD + DE + EF + FA$

Since $AB = BC = 3x + 5$, $CD = AB$, and $EF = 3x + 21$, $AF = DE = 3x + 2$, the total perimeter can be expressed as:

$\text{Perimeter} = (3x + 5) + (3x + 5) + (3x + 5) + (3x + 2) + (3x + 2) + (3x + 21)$

Now, simplify the expression:

$\text{Perimeter} = 3(3x + 5) + 2(3x + 2) + (3x + 21)$

$\text{Perimeter} = (9x + 15) + (6x + 4) + (3x + 21)$

$\text{Perimeter} = 18x + 40$

We are told that the perimeter is 172 meters, so:

$18x + 40 = 172$

Now, solve for $x$:

$18x = 172 - 40$

$18x = 132$

$x = \frac{132}{18} = 7.33 \, (\text{to 2 decimal places})$

Conclusion:

The value of $x$ is approximately $7.33$.

Would you like any further details or clarifications?

Here are 5 related questions you can explore:

1. What is the length of each side of the hexagon using this value of $x$?
2. How would the problem change if the perimeter was different, say 200 meters?
3. Can we verify the result geometrically, based on the trapezium's properties?
4. What would happen if the perpendicular distance between $BC$ and $FE$ changed?
5. How can we extend this to a similar hexagonal figure with different side relationships?

Tip: Always check that all conditions in the problem are used when solving geometry-based equations.