The centre of gravity of a trapezoidal dam section whose top width is a, bottom width is b and the vertical side is a, from its vertical face is A. $$\frac{{{{\text{a}}^2} + {\text{ab}} + {{\text{b}}^2}}}{{3\left( {{\text{a}} + {\text{b}}} \right)}}$$ B. $$\frac{{{{\text{b}}^2} + {\text{bc}} + {{\text{c}}^2}}}{{3\left( {{\text{b}} + {\text{c}}} \right)}}$$ C. $$\frac{{{{\text{a}}^2} + {\text{ac}} + {{\text{c}}^2}}}{{3\left( {{\text{a}} + {\text{c}}} \right)}}$$ D. None of these

The correct answer is $\frac{{{{\text{a}}^2} + {\text{ab}} + {{\text{b}}^2}}}{{3\left( {{\text{a}} + {\text{b}}} \right)}}$.

The center of gravity of a trapezoid is located at a point that is two-thirds of the way from the bottom to the top, and one-third of the way from the left to the right. In this case, the top width is $a$, the bottom width is $b$, and the vertical side is $a$. Therefore, the center of gravity is located at a point that is $\frac{2}{3}a$ from the bottom and $\frac{1}{3}a$ from the left. The distance from the center of gravity to the vertical face is then $\frac{2}{3}a – a = -\frac{a}{3}$. The area of the trapezoid is $\frac{1}{2}ab$. Therefore, the center of gravity of the trapezoidal dam section is located at a point that is $\frac{{{{\text{a}}^2} + {\text{ab}} + {{\text{b}}^2}}}{{3\left( {{\text{a}} + {\text{b}}} \right)}}$ from its vertical face.

Option A is incorrect because it does not take into account the fact that the center of gravity is located at a point that is $\frac{2}{3}a$ from the bottom and $\frac{1}{3}a$ from the left. Option B is incorrect because it does not take into account the fact that the area of the trapezoid is $\frac{1}{2}ab$. Option C is incorrect because it does not take into account the fact that the center of gravity is located at a point that is $\frac{a}{3}$ from the vertical face.