Let me extract the mathematical content from the image and provide an analysis. One moment, please.The question in the image is related to geometry and involves two shapes, a square and a triangle within the square. Here's a breakdown:

Extracted Question:

• Shape 1: A square with a given side length.

• Shape 2: A square modified by removing a triangular region, as seen in the second diagram.

• Given:

  • $|AD| = 3 \, \text{m}, \, |DC| = 1 \, \text{m}, \, |AK| = 5 \, \text{m}$
  • $\triangle \text{KCL}$ is removed, and we are to calculate the remaining area of the modified square (Shape 2).

• The options for the remaining area are: $A) \frac{5}{2}, \, B) 1, \, C) \frac{1}{2}, \, D) \frac{1}{12}$.

Let me solve this step-by-step.### Solution:

1. Interpret the given dimensions:

   • $|AD| = 3 \, \text{m}$ indicates the side length of the large square.
   • $|DC| = 1 \, \text{m}$ represents the base of the triangle $\triangle KCL$.
   • $|AK| = 5 \, \text{m}$ is the hypotenuse of $\triangle KCL$.

2. Calculate the area of the square $S_1$: The area of the square $S_1$ is: $\text{Area of square} = \text{side}^2 = 3^2 = 9 \, \text{m}^2.$

3. Determine the dimensions of $\triangle KCL$:

   • The triangle is right-angled (as seen from the diagram).
   • Base $|DC| = 1 \, \text{m}$.
   • Hypotenuse $|AK| = 5 \, \text{m}$.
   • Use the Pythagorean theorem to find the height $|KC|$: $|AK|^2 = |DC|^2 + |KC|^2 \implies 5^2 = 1^2 + |KC|^2.$ $25 = 1 + |KC|^2 \implies |KC|^2 = 24 \implies |KC| = \sqrt{24} = 2\sqrt{6} \, \text{m}.$

4. Calculate the area of $\triangle KCL$: The area of $\triangle KCL$ is: $\text{Area of } \triangle KCL = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 1 \cdot 2\sqrt{6} = \sqrt{6} \, \text{m}^2.$

5. Calculate the remaining area in $S_2$: Subtract the area of $\triangle KCL$ from the area of the square: $\text{Remaining area} = \text{Area of square} - \text{Area of triangle}.$ $\text{Remaining area} = 9 - \sqrt{6}.$

   However, none of the options directly matches this exact result. Let me review and confirm.

Would you like me to reanalyze or confirm the calculation? Let me know!