In triangle ABC, medians BD and CE intersect at F such that m∠BFE = 30°. If the lengths of BD and CE are 18 cm and 15 cm respectively, find the area, in square cm, of ∆ABC. 


Given triangle \( \Delta ABC \), medians \( BD \) and \( CE \) intersect at point \( F \). We know the following:

- \( m\angle BFE = 30^\circ \)
- Length of median \( BD = 18 \) cm
- Length of median \( CE = 15 \) cm

We need to find the area of \( \Delta ABC \).

### Step 1: Use the properties of the centroid and medians
The point \( F \) is the centroid of the triangle \( \Delta ABC \), and it divides each median in the ratio \( 2:1 \). Therefore, the lengths of \( BF \) and \( CF \) are:
\[
BF = \frac{2}{3} \times BD = \frac{2}{3} \times 18 = 12 \text{ cm}
\]
\[
CF = \frac{2}{3} \times CE = \frac{2}{3} \times 15 = 10 \text{ cm}
\]

### Step 2: Apply the formula for the area of the triangle using medians
The area \( A \) of triangle \( \Delta ABC \) can be calculated using the formula:
\[
\text{Area of } \Delta ABC = \frac{4}{3} \times \text{Area of } \Delta BFC
\]

### Step 3: Calculate the area of triangle \( \Delta BFC \)
\( \Delta BFC \) can be calculated using the formula for the area of a triangle when two sides and the included angle are known:
\[
\text{Area of } \Delta BFC = \frac{1}{2} \times BF \times CF \times \sin(\angle BFC)
\]
Given \( \angle BFE = 30^\circ \), and assuming \( \angle BFC = \angle BFE = 30^\circ \):
\[
\text{Area of } \Delta BFC = \frac{1}{2} \times 12 \times 10 \times \sin 30^\circ
\]
\[
\text{Area of } \Delta BFC = \frac{1}{2} \times 12 \times 10 \times \frac{1}{2} = 30 \text{ square cm}
\]

### Step 4: Calculate the area of \( \Delta ABC \)
Now, using the relation for the area of \( \Delta ABC \):
\[
\text{Area of } \Delta ABC = \frac{4}{3} \times 30 = 40 \text{ square cm}
\]

### Conclusion:
The area of \( \Delta ABC \) is **40 square cm**.

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Do you want details or have any questions?

Here are five related questions:
1. How would the area change if \( \angle BFE \) were \( 60^\circ \) instead of \( 30^\circ \)?
2. What is the significance of the centroid in a triangle regarding area?
3. Can the area of a triangle be calculated using medians without knowing the angle between them?
4. How do we derive the formula for the area of a triangle using two sides and the included angle?
5. What if \( BD \) and \( CE \) were not medians—how would that affect the area calculation?

**Tip:** Remember that the centroid divides each median into a \( 2:1 \) ratio, which simplifies calculations involving medians and areas.

Calculate the area of triangle ABC using given medians BD = 18 cm and CE = 15 cm, with an angle BFE = 30°. Understand how centroids and medians intersect at point F affect the triangle's area. Step-by-step solution included.

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