A quadrilateral has one vertex on each side of a square of side-length 1. If the lengths of the quadrilateral are a, b, c, d, then

Options

A. `2 le a^2 + b^2 + c^2 +d^2 le 4`
B. `1 le a^2 + b^2 + c^2 +d^2 le 4`
C. `sqrt 2 le a^2 + b^2 + c^2 + d^2 le 4`
D. `2 sqrt 2 le a^2 + b^2 + c^2 + d^2 le 4`

Answer

To solve the problem, we need to analyze the quadrilateral formed by placing one vertex on each side of a square with a side length of 1. We will denote the lengths of the sides of the quadrilateral as \( a, b, c, d \).

### Step-by-Step Solution:

1. **Understanding the Square**:
- We have a square with each side measuring 1 unit. Let's denote the vertices of the square as \( A, B, C, D \) in clockwise order.

2. **Placing the Vertices**:
- We place one vertex of the quadrilateral on each side of the square. Let’s denote these vertices as \( P, Q, R, S \) where:
- \( P \) is on side \( AB \)
- \( Q \) is on side \( BC \)
- \( R \) is on side \( CD \)
- \( S \) is on side \( DA \)

3. **Expressing Side Lengths**:
- The lengths of the sides of the quadrilateral can be expressed in terms of the distances between these points:
- \( a = |PQ| \)
- \( b = |QR| \)
- \( c = |RS| \)
- \( d = |SP| \)

4. **Using the Pythagorean Theorem**:
- Since the square's sides are perpendicular, we can use the Pythagorean theorem to express the lengths of the sides of the quadrilateral:
- For example, if \( P \) is at \( (x_1, 0) \) on side \( AB \), \( Q \) at \( (1, y_2) \) on side \( BC \), \( R \) at \( (x_3, 1) \) on side \( CD \), and \( S \) at \( (0, y_4) \) on side \( DA \), we can calculate:
- \( a = \sqrt{(x_1 - 1)^2 + (0 - y_2)^2} \)
- \( b = \sqrt{(1 - x_3)^2 + (y_2 - 1)^2} \)
- \( c = \sqrt{(x_3 - 0)^2 + (1 - y_4)^2} \)
- \( d = \sqrt{(0 - x_1)^2 + (y_4 - 0)^2} \)

5. **Finding the Sum of the Sides**:
- The total length of the quadrilateral is given by \( S = a + b + c + d \).

6. **Applying the Cauchy-Schwarz Inequality**:
- By applying the Cauchy-Schwarz inequality, we can establish bounds for \( S^2 \):
\[ S^2 = (a + b + c + d)^2 \leq (1 + 1 + 1 + 1)(a^2 + b^2 + c^2 + d^2) = 4(a^2 + b^2 + c^2 + d^2) \]
- Since the maximum value of \( a^2 + b^2 + c^2 + d^2 \) occurs when the vertices are at the corners of the square, we can conclude that \( S^2 \) is bounded between 2 and 4.

7. **Conclusion**:
- Thus, the sum of the lengths of the sides of the quadrilateral satisfies:
\[ 2 \leq S^2 \leq 4 \]
- Therefore, the sum \( S \) must be between \( \sqrt{2} \) and \( 2 \).

### Final Result:
The sum of the lengths of the quadrilateral \( a + b + c + d \) is constrained by the inequalities derived from the geometry of the square.