Did I Misuse the Parallel Axis Theorem for Polar Moment of Inertia?

[Usaid3112]

Homework Statement

the centroidal moment of inertia about the x axis for the area shown is 142.41 cm^4. Most nearly what is the centroidal polar moment of inertia. The answer is C. What confuses me is that the solution says that the Y axis passes through the Centroid of the shape so parallel axis theorem shouldn't be applicable. Clarifications would be appreciated.

Relevant Equations

Iyc= Iy + Ad^2

 

[Usaid3112]

The apparent correct answer given in the textbook is J=330 cm^4. I ended up using parallel axis theorem as I could not see how the Y axis passed through the centroid as the given y axis is completely towards the left side of the whole shape.My attempt at this question:-

 

[TSny]

The apparent correct answer given in the textbook is J=330 cm^4. I ended up using parallel axis theorem as I could not see how the Y axis passed through the centroid as the given y axis is completely towards the left side of the whole shape.My attempt at this question:-

The centroidal moment of inertia "about the x-axis" is given to be I_XC = 142.41 cm⁴. You can check that this is the moment of inertia about an axis parallel to the x-axis that passes through the centroid. So, it is not the moment of inertia about the x-axis as drawn in the figure.

The same applies to I_YC. So, you would not use the parallel axis theorem in calculating I_YC.