In `Delta ABC`, if lengths of medians BE and CF are 12 and 9 respectively, find the maximum value of `Delta`

Explanation

To find the maximum area of triangle ABC given the lengths of medians BE and CF, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of median BE = 12
- Length of median CF = 9

2. **Draw the Triangle and Medians:**
- Draw triangle ABC and label the vertices A, B, and C.
- Draw the medians BE and CF, where E is the midpoint of AC and F is the midpoint of AB.

3. **Locate the Centroid:**
- Let G be the centroid of triangle ABC. The centroid divides each median in the ratio 2:1.

4. **Calculate the Lengths from the Centroid:**
- For median BE:
\[ BG = \frac{2}{3} \times BE = \frac{2}{3} \times 12 = 8 \]
- For median CF:
\[ CG = \frac{2}{3} \times CF = \frac{2}{3} \times 9 = 6 \]

5. **Calculate the Area of Triangle BGC:**
- The area of triangle BGC can be calculated using the formula:
\[ \text{Area}_{BCG} = \frac{1}{2} \times BG \times CG \times \sin(\theta) \]
- Substituting the values:
\[ \text{Area}_{BCG} = \frac{1}{2} \times 8 \times 6 \times \sin(\theta) = 24 \sin(\theta) \]

6. **Calculate the Area of Triangle ABC:**
- The area of triangle ABC is three times the area of triangle BGC:
\[ \text{Area}_{ABC} = 3 \times \text{Area}_{BCG} = 3 \times 24 \sin(\theta) = 72 \sin(\theta) \]

7. **Find the Maximum Area:**
- The maximum value of \(\sin(\theta)\) is 1. Therefore, to find the maximum area:
\[ \text{Maximum Area}_{ABC} = 72 \times 1 = 72 \text{ square units} \]

### Final Answer:
The maximum area of triangle ABC is **72 square units**.

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