We are tasked with finding the length of $RU$, given that $PQRS$ and $STUV$ are squares, and their areas are $25 \, \text{cm}^2$ and $225 \, \text{cm}^2$, respectively.

Step 1: Side length of square $PQRS$

The area of square $PQRS$ is $25 \, \text{cm}^2$. Let the side length of this square be $x$. Then: $x^2 = 25 \implies x = 5 \, \text{cm}.$ Thus, the side length of square $PQRS$ is $5 \, \text{cm}$.

Step 2: Side length of square $STUV$

The area of square $STUV$ is $225 \, \text{cm}^2$. Let the side length of this square be $y$. Then: $y^2 = 225 \implies y = 15 \, \text{cm}.$ Thus, the side length of square $STUV$ is $15 \, \text{cm}$.

Step 3: Length of $RU$

Since $PQRS$ and $STUV$ are squares and $RST$, $RWU$, and $SWPV$ are straight lines, $RU$ is the diagonal of the larger square $STUV$. The diagonal $d$ of a square with side length $y$ is given by: $d = y\sqrt{2}.$ Substituting $y = 15 \, \text{cm}$: $RU = 15\sqrt{2} \, \text{cm}.$

Final Answer:

$RU = 15\sqrt{2} \, \text{cm} \approx 21.21 \, \text{cm}.$

Would you like a detailed diagram explanation or have further questions?

Here are some related questions you might explore:

1. What is the perimeter of square $STUV$?
2. What is the perimeter of square $PQRS$?
3. How does the diagonal of a square relate to its side length?
4. What is the difference between the areas of the two squares?
5. What is the ratio of the diagonals of $PQRS$ and $STUV$?

Tip: When solving for diagonals of squares, always remember they are scaled by ( \sqrt{2