The magnification \( M \) for a convex lens is calculated using:
\[\nM = \frac{\text{image height}}{\text{object height}} = \frac{v}{u}\n\]
Where:
- \( M = \frac{1}{2} \) (image size is half the object size)
- \( u \) is object distance, and \( v \) is image distance.
The lens equation is:
\[\n\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\n\]
With \( f = 12 \, \text{cm} \) (focal length).
From the magnification equation:
\[\n\frac{1}{2} = \frac{v}{u}\n\]
Therefore:
\[\nv = \frac{u}{2}\n\]
Substituting into the lens equation:
\[\n\frac{1}{12} = \frac{2}{u} - \frac{1}{u}\n\]
Solving for \( u \):
\[\n\frac{1}{12} = \frac{1}{u}\n\]
\[\nu = 36 \, \text{cm}\n\]
The object distance is 36 cm.