In a `triangle ABC`,AB = 10 cm, BC = 12 cm and AC = 14 cm. Find the length of median AD. If G is the centroid, find length of GA.

Options

A. `5/3sqrt7,5/9sqrt7`
B. `5sqrt7,4sqrt7`
C. `10/sqrt3,8/3sqrt7`
D. `4sqrt7,8/3sqrt7`

Answer

To find the length of the median AD in triangle ABC, where AB = 10 cm, BC = 12 cm, and AC = 14 cm, we can use the median formula. The median from vertex A to side BC (which is median AD) can be calculated using the following formula:

\[ m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \]

Where:
- \( m_a \) is the length of the median from vertex A to side BC,
- \( a \) is the length of side BC,
- \( b \) is the length of side AC,
- \( c \) is the length of side AB.

In our case:
- \( a = BC = 12 \, \text{cm} \)
- \( b = AC = 14 \, \text{cm} \)
- \( c = AB = 10 \, \text{cm} \)

Now, we can substitute the values into the formula:

\[ m_a = \frac{1}{2} \sqrt{2(14^2) + 2(10^2) - (12^2)} \]

Calculating each term step by step:

1. Calculate \( 14^2 \):
\[ 14^2 = 196 \]

2. Calculate \( 10^2 \):
\[ 10^2 = 100 \]

3. Calculate \( 12^2 \):
\[ 12^2 = 144 \]

4. Substitute these values back into the median formula:
\[ m_a = \frac{1}{2} \sqrt{2(196) + 2(100) - 144} \]

5. Calculate \( 2(196) \):
\[ 2(196) = 392 \]

6. Calculate \( 2(100) \):
\[ 2(100) = 200 \]

7. Now substitute these into the equation:
\[ m_a = \frac{1}{2} \sqrt{392 + 200 - 144} \]

8. Calculate \( 392 + 200 - 144 \):
\[ 392 + 200 = 592 \]
\[ 592 - 144 = 448 \]

9. Now we find the square root of 448:
\[ \sqrt{448} = \sqrt{16 \times 28} = 4\sqrt{28} = 4 \times 2\sqrt{7} = 8\sqrt{7} \]

10. Finally, substitute back to find \( m_a \):
\[ m_a = \frac{1}{2} \times 8\sqrt{7} = 4\sqrt{7} \, \text{cm} \]

Now, we have found the length of median AD:
\[ AD = 4\sqrt{7} \, \text{cm} \]

Next, we need to find the length of GA, where G is the centroid of triangle ABC. The centroid divides each median in the ratio 2:1. Therefore, the length of GA can be calculated as follows:

\[ GA = \frac{2}{3} AD \]

Substituting the value of AD:
\[ GA = \frac{2}{3} \times 4\sqrt{7} = \frac{8\sqrt{7}}{3} \, \text{cm} \]

Thus, the final answers are:
- Length of median AD = \( 4\sqrt{7} \, \text{cm} \)
- Length of GA = \( \frac{8\sqrt{7}}{3} \, \text{cm} \)