If PA is the median of the triangle PQR and G be the centroid, then what is the ratio of `(PA + GA) : (PG-GA)`?

Options

A. `4:1`
B. `3:1`
C. `2:1`
D. `3:2 `

Answer

To solve the problem, we need to understand the properties of the median and the centroid in a triangle.

1. **Understanding the Median and Centroid**:
- In a triangle, a median is a line segment joining a vertex to the midpoint of the opposite side.
- The centroid (G) of a triangle divides each median into two segments: the segment from the vertex to the centroid (GA) and the segment from the centroid to the midpoint of the opposite side (PG).

2. **Ratio of Segments**:
- The centroid divides the median in the ratio 2:1. This means that if PA is the entire length of the median, then:
- PG (the segment from the centroid to the midpoint) is 2/3 of the median.
- GA (the segment from the vertex to the centroid) is 1/3 of the median.
- Therefore, if we let PA = 3x, then GA = x and PG = 2x.

3. **Finding the Expression**:
- We need to calculate the ratio of (PA + GA) to (PG - GA).
- Substitute the values:
- PA + GA = 3x + x = 4x
- PG - GA = 2x - x = x

4. **Calculating the Ratio**:
- Now, we can find the ratio:
\[ \text{Ratio} = \frac{PA + GA}{PG - GA} = \frac{4x}{x} = 4 \]

Thus, the ratio of (PA + GA) to (PG - GA) is 4:1.