In `DeltaABC,D` and E are the mid points of sides BC and AC respectively. If `AD=10.8` cm, BE `=14.4` cm and AD and BE intersect at G at a right angle then the area `("in cm"^(2))` of `DeltaABC` is :

Options

A. A)`103.68`
B. B)`56.76`
C. C)`80.64`
D. D)`53.76`

Answer

To find the area of triangle ABC given the midpoints D and E, and the lengths of the medians AD and BE, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of median AD = 10.8 cm
- Length of median BE = 14.4 cm
- AD and BE intersect at point G at a right angle.

2. **Understand the Properties of Medians:**
- In triangle ABC, D is the midpoint of side BC, and E is the midpoint of side AC.
- The medians AD and BE intersect at the centroid G of the triangle.

3. **Use the Area Formula for Triangle with Medians:**
- The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{2}{3} \times AD \times BE \]
- This formula applies when the medians intersect at a right angle.

4. **Substitute the Values:**
- Substitute the values of AD and BE into the formula:
\[ \text{Area} = \frac{2}{3} \times 10.8 \times 14.4 \]

5. **Calculate the Area:**
- First, calculate the product of the medians:
\[ 10.8 \times 14.4 = 155.52 \]
- Now, substitute this product into the area formula:
\[ \text{Area} = \frac{2}{3} \times 155.52 \]
- Calculate:
\[ \text{Area} = \frac{311.04}{3} = 103.68 \text{ cm}^2 \]

6. **Final Result:**
- The area of triangle ABC is \( 103.68 \text{ cm}^2 \).