The medians of a triangle ABC meet at G. If the area of the triangle be 120 sq cm, then the area of the triangle GBC is

Options

A. 30 sq.cm
B. 50 sq. cm
C. 20 sq.cm
D. 40 sq.cm

Answer

To find the area of triangle GBC, we start by understanding the properties of the centroid (G) of triangle ABC. The centroid divides the triangle into three smaller triangles (ABG, BGC, and AGC) of equal area.

### Step-by-Step Solution:

1. **Identify the Area of Triangle ABC**:
We are given that the area of triangle ABC is 120 sq cm.

2. **Understanding the Centroid**:
The centroid (G) of a triangle divides it into three smaller triangles (ABG, BGC, AGC) that all have equal areas.

3. **Set Up the Equation**:
Since the areas of triangles ABG, BGC, and AGC are equal, we can express the total area of triangle ABC as:
\[ \text{Area of } ABC = \text{Area of } ABG + \text{Area of } BGC + \text{Area of } AGC \]
Let the area of triangle GBC be denoted as \( A_{GBC} \). Then:
\[ A_{ABG} = A_{BGC} = A_{AGC} = A_{GBC} \]

4. **Express Total Area in Terms of GBC**:
Since the areas are equal, we can write:
\[ \text{Area of } ABC = A_{GBC} + A_{GBC} + A_{GBC} = 3 \times A_{GBC} \]

5. **Substitute the Known Area**:
We know that the area of triangle ABC is 120 sq cm, so:
\[ 120 = 3 \times A_{GBC} \]

6. **Solve for Area of Triangle GBC**:
To find \( A_{GBC} \), divide both sides by 3:
\[ A_{GBC} = \frac{120}{3} = 40 \text{ sq cm} \]

### Final Answer:
The area of triangle GBC is **40 sq cm**.

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