https://file.aiquickdraw.com/mathbot/file/1730485203266u9d7zall.jpg

To solve this problem, let's break down the information provided and use geometric properties:

1. **Given Data:**
   - \( ABCD \) is a square with side length \( AB = 6 \).
   - Point \( I \) is the midpoint of \( AB \).
   - \( O \) is the intersection of line segments \( DI \) and \( AC \).
   - We are asked to determine the area of the orange triangle \( \triangle AOC \).

2. **Step-by-Step Solution:**
   - Since \( ABCD \) is a square with side \( AB = 6 \), each side of the square is 6 units long.
   - The total area of square \( ABCD \) is:
     \[
     \text{Area of } ABCD = AB^2 = 6^2 = 36 \text{ square units}.
     \]

3. **Using Coordinates:**
   - Place \( A \) at \( (0, 0) \), \( B \) at \( (6, 0) \), \( D \) at \( (0, 6) \), and \( C \) at \( (6, 6) \).
   - Since \( I \) is the midpoint of \( AB \), the coordinates of \( I \) are \( (3, 0) \).

4. **Finding Point \( O \):**
   - \( O \) is the intersection point of \( DI \) and \( AC \).
   - **Equation of \( DI \):** Since \( D \) is at \( (0, 6) \) and \( I \) is at \( (3, 0) \), the slope of \( DI \) is:
     \[
     \text{slope of } DI = \frac{0 - 6}{3 - 0} = -2.
     \]
     The equation of line \( DI \) (using point-slope form with point \( D(0, 6) \)) is:
     \[
     y = -2x + 6.
     \]
   - **Equation of \( AC \):** Since \( A \) is at \( (0, 0) \) and \( C \) is at \( (6, 6) \), the slope of \( AC \) is 1. So the equation of \( AC \) is:
     \[
     y = x.
     \]
   - **Finding \( O \):** Set \( y = -2x + 6 \) equal to \( y = x \):
     \[
     x = -2x + 6 \Rightarrow 3x = 6 \Rightarrow x = 2.
     \]
     Substitute \( x = 2 \) back into \( y = x \) to get \( y = 2 \).
     Thus, \( O \) is at \( (2, 2) \).

5. **Area of Triangle \( \triangle AOC \):**
   - Use the formula for the area of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):
     \[
     \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|.
     \]
   - For \( \triangle AOC \), with points \( A(0, 0) \), \( O(2, 2) \), and \( C(6, 6) \):
     \[
     \text{Area of } \triangle AOC = \frac{1}{2} \left| 0(2 - 6) + 2(6 - 0) + 6(0 - 2) \right|.
     \]
     \[
     = \frac{1}{2} \left| 0 + 12 - 12 \right| = \frac{1}{2} \times 0 = 0.
     \]

---

### Final Answer:
The area of the orange part, \( \triangle AOC \), is **0 square units** (if it is a degenerate triangle).

Sur la figure ci-contre, ABCD est un carré tel que AB = 6, I est le milieu de [AB] et O est le point d’intersection de (DI) et [AC]. Déterminer l’aire de la partie orange.

Learn how to calculate the area of a triangle formed within a square using geometry and coordinate geometry. This problem involves determining the area of triangle AOC where ABCD is a square with side AB = 6, I is the midpoint of AB, and O is the intersection of DI and AC. Suitable for Grade 9-10 students.

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