Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that:
$\bullet$ The area of square $KLMN$ is $1$.
$\bullet$ The vertices of the triangle divide three sides of the square up into these ratios:
$KX : XL = 3 : 2$
$KY : YN = 4 : 1$
$NZ : ZM = 2 : 3$
What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale).
[img]https://cdn.artofproblemsolving.com/attachments/8/0/38e76709373ba02346515f9949ce4507ed4f8f.png[/img]
1. **Determine the coordinates of the vertices of triangle \( XYZ \):**
   - The square \( KLMN \) has an area of 1, so each side of the square has length 1.
   - Given the ratios:
     - \( KX : XL = 3 : 2 \), so \( X \) divides \( KL \) in the ratio \( 3:2 \). Therefore, \( X \) is at \( \left( \frac{3}{5}, 1 \right) \).
     - \( KY : YN = 4 : 1 \), so \( Y \) divides \( KN \) in the ratio \( 4:1 \). Therefore, \( Y \) is at \( \left( 0, \frac{1}{5} \right) \).
     - \( NZ : ZM = 2 : 3 \), so \( Z \) divides \( NM \) in the ratio \( 2:3 \). Therefore, \( Z \) is at \( \left( \frac{2}{5}, 0 \right) \).

2. **Apply the Shoelace Theorem to find the area of triangle \( XYZ \):**
   - The Shoelace Theorem states that for vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), the area \( A \) is given by:
     \[
     A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right|
     \]
   - Substituting the coordinates \( X = \left( \frac{3}{5}, 1 \right) \), \( Y = \left( 0, \frac{1}{5} \right) \), \( Z = \left( \frac{2}{5}, 0 \right) \):
     \[
     A = \frac{1}{2} \left| \frac{3}{5} \cdot \frac{1}{5} + 0 \cdot 0 + \frac{2}{5} \cdot 1 - \left( 1 \cdot 0 + \frac{1}{5} \cdot \frac{2}{5} + 0 \cdot \frac{3}{5} \right) \right|
     \]
     \[
     = \frac{1}{2} \left| \frac{3}{25} + 0 + \frac{2}{5} - \left( 0 + \frac{2}{25} + 0 \right) \right|
     \]
     \[
     = \frac{1}{2} \left| \frac{3}{25} + \frac{10}{25} - \frac{2}{25} \right|
     \]
     \[
     = \frac{1}{2} \left| \frac{11}{25} \right|
     \]
     \[
     = \frac{11}{50}
     \]

The final answer is \(\boxed{\frac{11}{50}}\).