The given image shows a square $STUV$ with diagonals intersecting at point $W$. The side length $ST$ (or any side of the square) is labeled as $15$. We are tasked to find the following lengths:

1. $VU$
2. $SU$
3. $TV$
4. $SW$

Solution:

Step 1: Understanding the Square

• All sides of a square are equal. Hence, $ST = VU = SU = TV = 15$.

Step 2: Diagonals of the Square

• The diagonals of a square are equal in length and bisect each other at $W$.

• The diagonal of a square can be calculated using the Pythagorean Theorem:
  $\text{Diagonal} = \sqrt{\text{Side}^2 + \text{Side}^2} = \sqrt{2 \cdot \text{Side}^2}.$ Substituting $\text{Side} = 15$:
  $\text{Diagonal} = \sqrt{2 \cdot 15^2} = \sqrt{2 \cdot 225} = \sqrt{450}.$

• Simplifying $\sqrt{450}$:
  $\text{Diagonal} = 15\sqrt{2}.$

Step 3: Length of $SW$

• Since the diagonals bisect each other at $W$, each segment of the diagonal (e.g., $SW$) is half the length of the full diagonal:
  $SW = \frac{\text{Diagonal}}{2} = \frac{15\sqrt{2}}{2}.$

Final Answers:

1. $VU = 15$
2. $SU = 15$
3. $TV = 15$
4. $SW = \frac{15\sqrt{2}}{2}$.

Would you like a step-by-step explanation of the diagonals or further clarification on any part? Let me know!

Related Questions:

1. How is the diagonal of a square calculated using its side length?
2. What is the relationship between diagonals in a square?
3. If $STUV$ were a rectangle instead of a square, how would the diagonal change?
4. How do you simplify expressions involving square roots like $\sqrt{450}$?
5. What properties do the diagonals of a square satisfy geometrically?

Tip: In geometry, always verify symmetry in shapes like squares and rectangles to simplify calculations.