Let $T$ be the triangle in the $xy$-plane with vertices $(0, 0)$, $(3, 0)$, and $\left(0, \frac32\right)$. Let $E$ be the ellipse inscribed in $T$ which meets each side of $T$ at its midpoint. Find the distance from the center of $E$ to $(0, 0)$.
1. **Identify the vertices of the triangle \( T \):**
   The vertices of the triangle \( T \) are given as \( (0, 0) \), \( (3, 0) \), and \( \left(0, \frac{3}{2}\right) \).

2. **Determine the centroid of the triangle:**
   The centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
   \[
   G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
   \]
   Substituting the coordinates of the vertices:
   \[
   G = \left( \frac{0 + 3 + 0}{3}, \frac{0 + 0 + \frac{3}{2}}{3} \right) = \left( \frac{3}{3}, \frac{\frac{3}{2}}{3} \right) = \left( 1, \frac{1}{2} \right)
   \]

3. **Recognize the type of ellipse:**
   The problem states that the ellipse \( E \) is inscribed in \( T \) and meets each side of \( T \) at its midpoint. This is known as the Steiner inellipse, which is always centered at the centroid of the triangle.

4. **Calculate the distance from the centroid to the origin:**
   The distance \( d \) from the centroid \( (1, \frac{1}{2}) \) to the origin \( (0, 0) \) is given by the Euclidean distance formula:
   \[
   d = \sqrt{(1 - 0)^2 + \left( \frac{1}{2} - 0 \right)^2} = \sqrt{1^2 + \left( \frac{1}{2} \right)^2} = \sqrt{1 + \frac{1}{4}} = \sqrt{\frac{4}{4} + \frac{1}{4}} = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}
   \]

The final answer is \(\boxed{\frac{\sqrt{5}}{2}}\).