Let's solve this step-by-step.

Step 1: Understand the perimeter and side relations

The perimeter of the hexagon is given as 162 cm. We know it has six sides, but they are not all equal. We are told that:

1. Each longer side measures 1 cm more than twice a shorter side.
2. The hexagon has three pairs of equal sides: three longer sides and three shorter sides.

Let's label:

• The length of each shorter side as $s$.
• The length of each longer side as $2s + 1$.

Step 2: Set up the perimeter equation

Since there are three shorter sides and three longer sides, the perimeter can be written as: $3s + 3(2s + 1) = 162$

Expanding and simplifying the equation: $3s + 6s + 3 = 162$ $9s + 3 = 162$ Subtracting 3 from both sides: $9s = 159$ Dividing by 9: $s = 17.67 \, \text{cm} \, (\text{approx.})$

Step 3: Find the length of the longer side (labeled $x$)

Using the relation for the longer side: $x = 2s + 1 = 2(17.67) + 1 = 35.34 + 1 = 36.34 \, \text{cm} \, (\text{approx.})$

Conclusion

The length of the side labeled $x$ is approximately 36.34 cm.

Would you like a further breakdown of this calculation or have any other questions?

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Here are five related questions you might find interesting:

1. How would the perimeter change if all six sides were equal?
2. How can we calculate the area of a regular hexagon with equal side lengths?
3. What is the difference in perimeter between this hexagon and one where all sides are equal?
4. How would we solve this problem if given a different perimeter?
5. Can we verify the perimeter formula with different hexagon configurations?

Tip: In problems with proportional sides, setting variables for unknown lengths helps form equations easily!