Question: 9. BIRDS Sunil is designing a triangular-shaped bird bath. A pole will attach under the bird bath to hold it up.
Where should the pole be placed in order to balance the bird bath?
1. **State the problem:** We need to find the point where the pole should be placed to balance the triangular bird bath. This point is the centroid of the triangle formed by vertices $A(4, 5.33)$, $B(5.5, 4.83)$, and $C(4.76, 5.01)$.
2. **Formula for centroid:** The centroid $(x_g, y_g)$ of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by:
$$
(x_g, y_g) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
$$
3. **Apply the formula:** Substitute the coordinates:
$$
x_g = \frac{4 + 5.5 + 4.76}{3} = \frac{14.26}{3} = 4.7533\ldots
$$
$$
y_g = \frac{5.33 + 4.83 + 5.01}{3} = \frac{15.17}{3} = 5.0567\ldots
$$
4. **Interpretation:** The centroid is approximately at $(4.75, 5.06)$.
5. **Compare with given options:** The closest option to the centroid is $C(4.76, 5.01)$ and $D(5, 5)$. Among these, $D(5, 5)$ is a good approximation and is given as the center.
**Final answer:** The pole should be placed at the centroid, approximately at point $D(5, 5)$ to balance the bird bath.