Consider a triange ABC, where B and C are `(-a,0) and (a,0)` respectively . A be any point (h,k) P,Q,R divides the sides of this triangle in same ratio. Then centroid of `Delta PQR` is

Options

A. `(0,0)`
B. `((h)/(3), 0)`
C. `(0,(k)/(3))`
D. `((h)/(3), (k)/(3))`

Answer

To find the centroid of triangle PQR, we start with the given points of triangle ABC and the point A.

### Step-by-Step Solution:

1. **Identify the Points of Triangle ABC:**
- The points are given as:
- \( B = (-a, 0) \)
- \( C = (a, 0) \)
- Let \( A = (h, k) \)

2. **Determine the Ratios for Points P, Q, R:**
- Points P, Q, and R divide the sides of triangle ABC in the same ratio \( m:1 \).
- The coordinates of points P, Q, and R can be calculated using the section formula.

3. **Calculate the Coordinates of Point P:**
- Point P divides side AB in the ratio \( m:1 \):
\[ P = \left( \frac{-am + h}{m+1}, \frac{0 + k}{m+1} \right) = \left( \frac{-am + h}{m+1}, \frac{k}{m+1} \right) \]

4. **Calculate the Coordinates of Point Q:**
- Point Q divides side AC in the ratio \( m:1 \):
\[ Q = \left( \frac{am + h}{m+1}, \frac{0 + k}{m+1} \right) = \left( \frac{am + h}{m+1}, \frac{k}{m+1} \right) \]

5. **Calculate the Coordinates of Point R:**
- Point R divides side BC in the ratio \( m:1 \):
\[ R = \left( \frac{am - a}{m+1}, 0 \right) = \left( \frac{a(m-1)}{m+1}, 0 \right) \]

6. **Find the Centroid of Triangle PQR:**
- The centroid \( G \) of triangle PQR is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
- Substituting the coordinates of P, Q, and R:
\[ G_x = \frac{\frac{-am + h}{m+1} + \frac{am + h}{m+1} + \frac{a(m-1)}{m+1}}{3} \]
\[ G_y = \frac{\frac{k}{m+1} + \frac{k}{m+1} + 0}{3} \]

7. **Simplify the Centroid Coordinates:**
- For \( G_x \):
\[ G_x = \frac{\frac{-am + h + am + h + a(m-1)}{m+1}}{3} = \frac{\frac{2h + a(m-1)}{m+1}}{3} = \frac{2h + a(m-1)}{3(m+1)} \]
- For \( G_y \):
\[ G_y = \frac{2k}{3(m+1)} \]

8. **Final Result:**
- The centroid \( G \) of triangle PQR is:
\[ G = \left( \frac{h}{3}, \frac{k}{3} \right) \]