In ABC, G is the centroid, AB = 15 cm, BC = 18 cm and AC = 25 cm. Find GD, where D is the mid-point of BC.

Options

A. `1/3sqrt86` cm
B. `2/3sqrt86` cm
C. `8/3sqrt15`
D. none of these

Answer

To find the length GD in triangle ABC where G is the centroid and D is the midpoint of side BC, we can follow these steps:

### Step 1: Understand the triangle and its properties
- We have triangle ABC with sides AB = 15 cm, BC = 18 cm, and AC = 25 cm.
- D is the midpoint of BC, so BD = DC = 9 cm.

**Hint:** Remember that the centroid divides each median in a 2:1 ratio.

### Step 2: Calculate the lengths of the segments
- Since D is the midpoint of BC, we have:
- BD = DC = 9 cm.

### Step 3: Find the length of the median AD
- To find GD, we first need to calculate the length of the median AD using the formula for the length of a median:

\[ m_a = \sqrt{\frac{2b^2 + 2c^2 - a^2}{4}} \]

where \( a = BC \), \( b = AC \), and \( c = AB \).

Here, \( a = 18 \) cm, \( b = 25 \) cm, and \( c = 15 \) cm.

### Step 4: Substitute the values into the median formula
- Substitute the values into the median formula:

\[ m_a = \sqrt{\frac{2(25^2) + 2(15^2) - (18^2)}{4}} \]

Calculating each term:
- \( 25^2 = 625 \)
- \( 15^2 = 225 \)
- \( 18^2 = 324 \)

Now substituting these values:

\[ m_a = \sqrt{\frac{2(625) + 2(225) - 324}{4}} = \sqrt{\frac{1250 + 450 - 324}{4}} = \sqrt{\frac{1376}{4}} = \sqrt{344} \]

### Step 5: Calculate GD
- The centroid G divides the median AD in the ratio 2:1. Therefore, the length GD is:

\[ GD = \frac{1}{3} \times m_a = \frac{1}{3} \times \sqrt{344} \]

Calculating \( \sqrt{344} \):
- \( \sqrt{344} \approx 18.52 \) (approximately).

Thus,

\[ GD \approx \frac{18.52}{3} \approx 6.17 \text{ cm} \]

### Final Answer
GD is approximately \( 6.17 \) cm.

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