Medians of $\Delta ABC$ intersect at G. If area of $\Delta ABC=27~cm^{2},$ then area of $\Delta BGC=$

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to understand the properties of medians and centroids in a triangle.

1. **Understanding the Role of the Centroid:**
   - The centroid (G) of a triangle is the point where its three medians intersect. It divides each median into two segments, with the segment connecting the vertex to the centroid being twice as long as the segment connecting the centroid to the midpoint of the opposite side.
   - The centroid divides the triangle into three smaller triangles of equal area.

2. **Area Division by the Centroid:**
   - Given that the area of $\Delta ABC$ is $27 \, \text{cm}^2$, the centroid divides this area into three smaller triangles: $\Delta AGC$, $\Delta BGC$, and $\Delta CGA$.
   - Each of these smaller triangles has an equal area because the centroid divides the triangle into three triangles of equal area.

3. **Calculating the Area of $\Delta BGC$:**
   - Since the total area of $\Delta ABC$ is $27 \, \text{cm}^2$, and the centroid divides it into three triangles of equal area, each of these triangles, including $\Delta BGC$, has an area of:
     $$
     \text{Area of } \Delta BGC = \frac{27 \, \text{cm}^2}{3} = 9 \, \text{cm}^2
     $$

4. **Explanation of Incorrect Options:**
   - **Option 2: $6 \, \text{cm}^2$**: This option is incorrect because it suggests that the area of $\Delta BGC$ is less than one-third of the total area of $\Delta ABC$, which contradicts the property of the centroid dividing the triangle into three equal areas.
   - **Option 3: $12 \, \text{cm}^2$**: This option is incorrect because it implies that the area of $\Delta BGC$ is more than one-third of the total area, which is not possible given the equal division by the centroid.
   - **Option 4: $18 \, \text{cm}^2$**: This option is incorrect as it suggests that $\Delta BGC$ occupies two-thirds of the total area, which is not consistent with the equal division by the centroid.

In conclusion, the correct answer is Option 1: $9 \, \text{cm}^2$, as it accurately reflects the division of the triangle's area by the centroid into three equal parts.