https://file.aiquickdraw.com/mathbot/file/1726456425638uhgn7lvt.jpg

To solve the question, we need to determine the transformation matrix that represents the dilation of the square \( OABC \) to the rectangle \( OA'B'C' \).

### Step-by-Step Analysis

1. **Understanding Dilation:**
   A dilation transformation is a linear transformation that scales a figure by a constant factor along the x-axis and the y-axis. The scaling factor could be different for each axis.

2. **From the Diagram:**
   - The original square \( OABC \) has coordinates:
     - \( O(0, 0) \)
     - \( A(0, 3) \)
     - \( B(3, 3) \)
     - \( C(3, 0) \)
   - The image rectangle \( OA'B'C' \) has coordinates:
     - \( O'(0, 0) \)
     - \( A'(0, 2) \)
     - \( B'(3, 2) \)
     - \( C'(3, 0) \)

3. **Scaling Factors:**
   - Along the **y-axis**: The height of the original figure \( A \) is 3 units, and the height of the dilated figure \( A' \) is 2 units. Therefore, the dilation factor along the y-axis is \( \frac{2}{3} \).
   - Along the **x-axis**: The width of the original and dilated figure remains the same (both \( B \) and \( C \) are 3 units from the origin), so the dilation factor along the x-axis is 1.

4. **Transformation Matrix:**
   The transformation matrix for a dilation is represented as:
   \[
   \begin{pmatrix}
   \text{Scale factor along the x-axis} & 0 \\
   0 & \text{Scale factor along the y-axis}
   \end{pmatrix}
   \]
   Based on the scaling factors identified:
   \[
   \begin{pmatrix}
   1 & 0 \\
   0 & \frac{2}{3}
   \end{pmatrix}
   \]

5. **Correct Answer:**
   The matrix that matches this is option (C):
   \[
   \begin{pmatrix}
   1 & 0 \\
   0 & \frac{2}{3}
   \end{pmatrix}
   \]

Let me know if you'd like further clarifications or details!

### Related Questions:
1. What is the determinant of a dilation matrix, and how does it relate to area change?
2. How do you derive the dilation matrix for other transformations, such as reflections or rotations?
3. What are the properties of linear transformations like dilation?
4. Can dilation matrices represent negative scaling factors (reflections)?
5. How do dilation matrices behave when combined with other transformation matrices (e.g., rotations)?

### Tip:
When dealing with dilation, always check both axes independently to determine the appropriate scaling factors.

A dilation from the square OABC to its image rectangle OA'B'C' is shown in the diagram. What is the transformation matrix representing this dilation?

Learn how to find the transformation matrix for a dilation from the square OABC to the rectangle OA'B'C'. This problem covers key geometry and transformation concepts, with an emphasis on scaling factors and the dilation matrix. Suitable for high school students in Grades 10-12.

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