Let's break down the problem:

1. Given: The area of the rectangle is 90, and it can be cut into two squares.
2. Question: We need to find the length of each side of the squares, represented by an irrational number.

Solution:

Let $x$ be the length of the side of each square.

• The area of one square is $x^2$.
• Since the rectangle is cut into two squares, the total area of the rectangle will be the sum of the areas of the two squares, i.e., $2x^2$.

Given that the area of the rectangle is 90: $2x^2 = 90$

Dividing both sides by 2: $x^2 = 45$

Taking the square root of both sides: $x = \sqrt{45}$

Simplifying $\sqrt{45}$: $x = \sqrt{9 \times 5} = 3\sqrt{5}$

Thus, the length of each side of the squares is $3\sqrt{5}$, which is an irrational number.

Final Answer:

The length of each side of the squares is $3\sqrt{5}$, which is an irrational number.

Would you like further details or explanations? Here are five related questions to think about:

1. How do we know that $\sqrt{45}$ is irrational?
2. Can we approximate $3\sqrt{5}$ to a decimal value?
3. How would the solution change if the rectangle were cut into more than two squares?
4. What are other examples of irrational numbers that can represent lengths in geometric problems?
5. How do you determine if a square root simplifies to a rational number?

Tip: When simplifying square roots, always factor the number into its prime components to check if there are perfect squares you can simplify.