If `G_1 and G_2` are the centroid of `triangleABC and trianglePQR` respectively, then `overline(AP)+overline(BQ)+overline(CR)=`

Options

A. `overline(G_1G_2)`
B. `2overline(G_1G_2)`
C. `3overline(G_1G_2)`
D. `6overline(G_1G_2)`

Answer

To solve the problem, we need to find the value of \( \overline{AP} + \overline{BQ} + \overline{CR} \) given that \( G_1 \) and \( G_2 \) are the centroids of triangles \( ABC \) and \( PQR \) respectively.

### Step-by-Step Solution:

1. **Identify the Centroid Formulas**:
The position vector of the centroid \( G_1 \) of triangle \( ABC \) is given by:
\[ \overline{G_1} = \frac{\overline{A} + \overline{B} + \overline{C}}{3} \]
Similarly, the position vector of the centroid \( G_2 \) of triangle \( PQR \) is:
\[ \overline{G_2} = \frac{\overline{P} + \overline{Q} + \overline{R}}{3} \]

2. **Express Vectors \( \overline{AP}, \overline{BQ}, \overline{CR} \)**:
The vectors can be expressed as:
\[ \overline{AP} = \overline{P} - \overline{A} \]
\[ \overline{BQ} = \overline{Q} - \overline{B} \]
\[ \overline{CR} = \overline{R} - \overline{C} \]

3. **Combine the Vectors**:
Now, we can combine these vectors:
\[ \overline{AP} + \overline{BQ} + \overline{CR} = (\overline{P} - \overline{A}) + (\overline{Q} - \overline{B}) + (\overline{R} - \overline{C}) \]
Simplifying this gives:
\[ \overline{AP} + \overline{BQ} + \overline{CR} = (\overline{P} + \overline{Q} + \overline{R}) - (\overline{A} + \overline{B} + \overline{C}) \]

4. **Substitute Centroid Values**:
Using the centroid formulas, we can substitute:
\[ \overline{P} + \overline{Q} + \overline{R} = 3\overline{G_2} \]
\[ \overline{A} + \overline{B} + \overline{C} = 3\overline{G_1} \]
Therefore, we have:
\[ \overline{AP} + \overline{BQ} + \overline{CR} = 3\overline{G_2} - 3\overline{G_1} \]

5. **Factor Out the Common Term**:
We can factor out the 3:
\[ \overline{AP} + \overline{BQ} + \overline{CR} = 3(\overline{G_2} - \overline{G_1}) \]

6. **Final Result**:
Thus, the final result is:
\[ \overline{AP} + \overline{BQ} + \overline{CR} = 3 \overline{G_2 G_1} \]