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Buscar entre las figuras siguientes aquellas cuya área sea una función afín de a o una función lineal de a.

To solve this problem, we need to determine which of the given shapes have an area that is either an affine function or a linear function of the variable \( a \).

An affine function is of the form \( f(a) = ma + b \), where \( m \) and \( b \) are constants, and a linear function is a special case where \( b = 0 \).

### Analyzing Each Figure:

1. **Figure N (Rectangle)**:
   - The area of a rectangle is given by \( A = \text{base} \times \text{height} \).
   - Since both the base and height appear independent of \( a \), the area could be of the form \( ma + b \).

2. **Figure P (Square)**:
   - The area of a square is \( A = \text{side}^2 \).
   - If the side is a linear function of \( a \), say \( s = k_1 a + k_2 \), then \( A = (k_1 a + k_2)^2 \) which expands to a quadratic expression in \( a \). This is not affine or linear.

3. **Figure Q (Right Triangle)**:
   - The area of a right triangle is \( A = \frac{1}{2} \times \text{base} \times \text{height} \).
   - If either the base or height is a linear function of \( a \), the area will also be linear in \( a \).

4. **Figure M (Rectangle split in half)**:
   - The area of this figure would be half the area of a rectangle. Thus, similar to the analysis for figure N, this area could be linear or affine in \( a \) depending on the side lengths.

### Conclusion:
- **Figure N** and **Figure M** are the figures whose area could be affine functions of \( a \), as the area is dependent on the product of dimensions, and one of these dimensions might vary linearly with \( a \).

- **Figure Q** could also be an affine or linear function depending on how the base or height varies with \( a \).

### Final Answer:
Figures **N**, **M**, and **Q** have areas that could be either affine or linear functions of \( a \).

Would you like more detailed explanations for these calculations? Here are some related questions:

1. What are the differences between affine and linear functions?
2. How do you expand a quadratic expression to determine if it is linear or not?
3. Can you derive the area formula for different shapes using different approaches?
4. What conditions must be met for a geometric figure's area to be an affine function?
5. How does the linearity of the side length affect the area of common geometric shapes?

**Tip**: When determining if a function is affine or linear, always check how the variable appears—whether it is raised to a power or remains in its simplest form.

Descubre qué figuras geométricas tienen áreas que pueden ser funciones afines o lineales de la variable a. Explicación detallada con ejemplos de figuras como rectángulos, cuadrados y triángulos rectángulos.

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