G is the centroid of the equilateral `DeltaABC`. If `AB = 10 cm `then length of AG is

Options

A. `(5sqrt(3))/(3)` cm
B. `(10sqrt(3))/(3) cm`
C. `5sqrt(3)` cm
D. `10 sqrt(3)` cm

Answer

To find the length of AG in an equilateral triangle ABC where G is the centroid and AB = 10 cm, we can follow these steps:

### Step 1: Understand the properties of the triangle
Since ABC is an equilateral triangle, all sides are equal. Therefore, we have:
- AB = AC = BC = 10 cm.

### Step 2: Find the midpoint D of side BC
In an equilateral triangle, the centroid (G) divides each median in a 2:1 ratio. We need to find the length of the median AD. First, we find the midpoint D of side BC. Since BC = 10 cm, we have:
- BD = DC = 5 cm.

### Step 3: Use Apollonius's theorem
Apollonius's theorem states that in any triangle, the sum of the squares of any two sides is equal to twice the square of the median to the third side plus twice the square of half the third side. Therefore, we can write:
\[ AB^2 + AC^2 = 2AD^2 + 2BD^2. \]

Substituting the known values:
\[ 10^2 + 10^2 = 2AD^2 + 2(5^2). \]
\[ 100 + 100 = 2AD^2 + 2(25). \]
\[ 200 = 2AD^2 + 50. \]

### Step 4: Solve for AD
Now, we can solve for AD:
\[ 200 - 50 = 2AD^2. \]
\[ 150 = 2AD^2. \]
\[ AD^2 = \frac{150}{2} = 75. \]
\[ AD = \sqrt{75} = 5\sqrt{3} \text{ cm}. \]

### Step 5: Find AG
Since G divides AD in the ratio 2:1, we can find AG:
\[ AG = \frac{2}{3} AD = \frac{2}{3} \times 5\sqrt{3} = \frac{10\sqrt{3}}{3} \text{ cm}. \]

Thus, the length of AG is:
\[ AG = \frac{10\sqrt{3}}{3} \text{ cm}. \]

### Final Answer
The length of AG is \( \frac{10\sqrt{3}}{3} \) cm.

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