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El área sombreada es el área con puntitos

The problem asks for the area of the shaded region within a square \( ABCD \) where \( AB = BC = CD = DA = 30 \) meters. Points \( M \) and \( N \) are midpoints of sides \( AD \) and \( AB \), respectively.

### Step-by-Step Solution:

1. **Determine the midpoints \( M \) and \( N \):**
   - Since \( M \) and \( N \) are midpoints, \( AM = MD = 15 \) meters and \( AN = NB = 15 \) meters.

2. **Identify the triangle \( BMN \) and the quadrilateral \( MNCD \):**
   - The line \( MN \) is a diagonal of the triangle \( BMN \), which splits it into two right-angled triangles.
   - The triangle \( BMN \) is right-angled at \( N \) because \( AB \) and \( MN \) are perpendicular.

3. **Calculate the area of triangle \( BMN \):**
   - The area of triangle \( BMN \) can be found using:
   \[
   \text{Area of } \triangle BMN = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AN \times BM = \frac{1}{2} \times 15 \times 15 = 112.5 \text{ square meters.}
   \]

4. **Calculate the area of the square \( ABCD \):**
   - The area of the square \( ABCD \) is:
   \[
   \text{Area of } ABCD = 30 \times 30 = 900 \text{ square meters.}
   \]

5. **Calculate the area of quadrilateral \( MNCD \):**
   - The quadrilateral \( MNCD \) is formed by subtracting the area of triangle \( BMN \) from the area of triangle \( ABC \), and since \( MN \) cuts through \( AC \) (which is the diagonal and divides the square into two equal parts), the area of \( MNCD \) is equal to the area of the triangle \( BMN \), i.e., 112.5 square meters.

6. **Find the area of the shaded region:**
   - The shaded region is a triangle, which is half of the triangle \( BMN \) that was calculated before.
   - Since triangle \( BMN \) is isosceles, its shaded half is:
   \[
   \text{Area of the shaded region} = \frac{1}{2} \times \text{Area of } \triangle BMN = \frac{1}{2} \times 112.5 = 56.25 \text{ square meters.}
   \]

Thus, the area of the shaded region is **56.25 square meters**.

Would you like more details, or do you have any questions?

### Related Questions:

1. How would the area of the shaded region change if \( M \) and \( N \) were one-third points instead of midpoints?
2. What is the area of the triangle \( ABC \) without the line \( MN \)?
3. How do you calculate the area of the square \( ABCD \) using the diagonal?
4. Can you generalize the shaded area formula if the square has side length \( s \)?
5. What if the square was rotated by 45 degrees? How would that affect the shaded area?

### Tip:

When dealing with symmetrical shapes, exploiting the symmetry can simplify calculations considerably, as seen in this problem.

Learn how to calculate the area of the shaded region within a square ABCD where AB = BC = CD = DA = 30 meters. Step-by-step solution provided, including the use of triangle properties and area calculation formulas. Suitable for students in Grades 7-9.

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