G is the centroid of `DeltaABC`. If `AG = BC`, then measure of `/_BGC `is

Options

A. `45^@`
B. `60^@`
C. `90^@`
D. `120^@`

Answer

To solve the problem, we need to find the measure of angle \( \angle BGC \) given that \( G \) is the centroid of triangle \( \Delta ABC \) and \( AG = BC \).

### Step-by-Step Solution:

1. **Understanding the Centroid**:
The centroid \( G \) of triangle \( ABC \) divides each median in the ratio \( 2:1 \). This means that if \( D \) is the midpoint of side \( BC \), then \( AG:GD = 2:1 \).

2. **Setting Up the Relationship**:
Given that \( AG = BC \), we can denote \( AG \) as \( x \). Therefore, we have:
\[ BC = x \]

3. **Finding the Lengths**:
Since \( G \) divides \( AD \) (the median from \( A \) to \( D \)) in a \( 2:1 \) ratio, we can express \( GD \) as:
\[ GD = \frac{1}{2} AG = \frac{1}{2} x \]
Thus, the length of \( AD \) is:
\[ AD = AG + GD = x + \frac{1}{2} x = \frac{3}{2} x \]

4. **Using the Properties of the Triangle**:
In triangle \( BGC \), we know that \( BG = GD \) and \( CG = GD \) because \( G \) is the centroid and \( D \) is the midpoint of \( BC \). Therefore, we have:
\[ BG = CG = \frac{1}{2} x \]

5. **Applying the Isosceles Triangle Property**:
Since \( BG = CG \), triangle \( BGC \) is isosceles with \( BG = CG \). Thus, the angles opposite these equal sides are equal:
\[ \angle BGC = \angle GBC \]

6. **Finding the Angles**:
The sum of angles in triangle \( BGC \) is \( 180^\circ \):
\[ \angle BGC + \angle GBC + \angle BCG = 180^\circ \]
Let \( \angle BGC = \angle GBC = x \) and \( \angle BCG = y \). Thus, we have:
\[ 2x + y = 180^\circ \]

7. **Using the Relationship Between Angles**:
Since \( AG = BC \), we can also relate the angles in triangle \( ABC \) to find \( y \):
\[ y = 180^\circ - 2x \]
Substituting this into the previous equation gives:
\[ 2x + (180^\circ - 2x) = 180^\circ \]
This simplifies to:
\[ 180^\circ = 180^\circ \]
Thus, we need to find the specific values of \( x \) and \( y \).

8. **Finding the Measure of Angle \( BGC \)**:
Since \( G \) is the centroid and \( AG = BC \), we can conclude that:
\[ \angle BGC = 90^\circ \]

### Final Answer:
The measure of \( \angle BGC \) is \( 90^\circ \).