Let's solve the problem.

The perimeter of quadrilateral $J K L M$ is given as 90 cm. The sides are labeled in the diagram as follows:

• $J K = n + 7$
• $K L = n$
• $L M = 2n - 12$
• $M J = 3n + 4$

The perimeter is the sum of all the sides, so we have: $(n + 7) + n + (2n - 12) + (3n + 4) = 90$

Now, simplify the expression: $n + 7 + n + 2n - 12 + 3n + 4 = 90$ $7n - 1 = 90$ Add 1 to both sides: $7n = 91$ Solve for $n$: $n = \frac{91}{7} = 13$

Now that we know $n = 13$, we can find the length of side $K L$: $K L = n = 13 \, \text{cm}$

Thus, the length of side $K L$ is 13 cm.

Would you like more details or explanations? Here are five related questions to expand your understanding:

1. What is the length of side $J K$ when $n = 13$?
2. What is the length of side $L M$ when $n = 13$?
3. What is the length of side $M J$ when $n = 13$?
4. How does changing the perimeter affect the value of $n$?
5. How do we verify that the total perimeter is 90 cm after calculating all sides?

Tip: Always double-check each part of your equation after solving for $n$ to ensure the calculations are correct.