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.