A uniform plane sheet of metal in the form of a triangle `ABC` has `BC gt AB gt AC`. Its moment of inertia will be smallest

Options

A. about `AC` as axis
B. about `AB` as axis
C. about `BC` as axis
D. with a line through `C` normal to its plane as axis,

Explanation

To determine the moment of inertia of a uniform plane sheet of metal in the form of triangle ABC, where BC > AB > AC, we need to analyze the distribution of mass relative to the axes of rotation.

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia (I) of a body depends on how the mass is distributed with respect to the axis of rotation. It is calculated using the formula:
\[ I = \int r^2 \, dm \]
where \( r \) is the distance from the axis of rotation to the mass element \( dm \).

2. **Identifying the Triangle**:
We have triangle ABC with sides such that \( BC > AB > AC \). This means that side BC is the longest, followed by AB, and AC is the shortest.

3. **Choosing the Axis of Rotation**:
To find which axis gives the smallest moment of inertia, we need to consider the axes along each side of the triangle. We will analyze the moment of inertia about the sides BC, AB, and AC.

4. **Analyzing Mass Distribution**:
- **About Side BC**: The distance of the mass elements from the axis (BC) will be the shortest for the other two sides (AB and AC). This means that the mass is distributed closer to the axis, resulting in a smaller moment of inertia.
- **About Side AB**: The distance from the axis (AB) to the mass elements (AC and BC) will be greater compared to the axis BC, leading to a larger moment of inertia.
- **About Side AC**: Similarly, the distance from the axis (AC) to the mass elements (AB and BC) will also be larger than that for axis BC, resulting in a larger moment of inertia.

5. **Conclusion**:
Since the mass distribution is closest to the axis when rotating about side BC, the moment of inertia will be smallest about this axis. Thus, the moment of inertia of the triangle ABC will be smallest when calculated about side BC.

### Final Answer:
The moment of inertia of the uniform plane sheet of metal in the form of triangle ABC will be smallest about side BC.