In `Delta ABC, G` is the centroid , AB=15cm, BC=18 cm, and AC=25 cm . Find GD , Where D is the mid point of BC:

Answer

To solve the problem step by step, we will follow the process outlined in the video transcript.

### Step 1: Understand the Problem
We have a triangle ABC with sides AB = 15 cm, BC = 18 cm, and AC = 25 cm. We need to find the length of GD, where G is the centroid of the triangle and D is the midpoint of side BC.

### Step 2: Find the Length of AD (Median)
To find GD, we first need to calculate the length of the median AD. We can use Apollonius's theorem, which states:

\[ AB^2 + AC^2 = 2AD^2 + \left(\frac{BC}{2}\right)^2 \]

Substituting the given values:

- \(AB = 15\) cm
- \(AC = 25\) cm
- \(BC = 18\) cm

Calculating:

\[ 15^2 + 25^2 = 2AD^2 + \left(\frac{18}{2}\right)^2 \]

Calculating the squares:

\[ 225 + 625 = 2AD^2 + 9^2 \]

\[ 225 + 625 = 2AD^2 + 81 \]

\[ 850 = 2AD^2 + 81 \]

### Step 3: Solve for AD^2
Now, isolate \(AD^2\):

\[ 850 - 81 = 2AD^2 \]

\[ 769 = 2AD^2 \]

\[ AD^2 = \frac{769}{2} \]

### Step 4: Calculate AD
Now, take the square root to find AD:

\[ AD = \sqrt{\frac{769}{2}} = \frac{\sqrt{769}}{\sqrt{2}} = \frac{\sqrt{769 \cdot 2}}{2} = \frac{\sqrt{1538}}{2} \]

### Step 5: Find GD
The centroid G divides the median AD in the ratio 2:1. Therefore, we can express GD as:

\[ GD = \frac{1}{3}AD \]

Substituting the value of AD:

\[ GD = \frac{1}{3} \cdot \frac{\sqrt{1538}}{2} \]

\[ GD = \frac{\sqrt{1538}}{6} \]

### Final Answer
Thus, the length of GD is:

\[ GD = \frac{\sqrt{1538}}{6} \text{ cm} \]